Yes. A really close fit, fulfilling all of your criteria in one narrative is the origin of a Burmese king called Pyusawhti, who is hatched from a dragon's egg and raised by peasants. Pyusawhti has a sister, hatched from a different egg, but who grows up separately from him. [For the whole story, see the section A Dragon Princess and the Kingdoms of Burma below.]
Numerous other stories bear similar elements, but none correspond quite as closely to your list of requirements. Another Greek tale parallels Leda's specifically in the versions thereof in which there is no goddess involved and Leda herself is the one said to have laid an egg (or two) after consorting with Zeus. [See the section Molione and Her Twin Sons below.] There is at least one Inca parallel to these Greek myths. [See A Quintuple Deity, & Twin Gods, Among the Inca below.]
Alternatively, in some other South East Asian narratives, there are divinities who lay eggs from which heroes hatch but then these children are raised by their own parents and not by humans. Yet a different motif occurs in an Inca creation myth in which the sun-god sends to his son Vichama three eggs, each made of a different metal, out of which society's different strata are hatched.
Very similarly, in the Nuhiño, "Earth Story," of the Jivaro of Ecuador and Peru: Chingaso, the wife of the creator-god Kumpara, gives to her son Etsa (the Sun) and to Etsa's wife Nantu (the Moon) an egg from which the first human woman, Mika, hatches. The birds and animals of the forests are later produced from other eggs given to Nantu by Chingaso.
On pp. 36-37 of her Thematic Guide to World Mythology, Lorena Stookey says that:
In a tale from the Admiralty Island... people are born from the
eggs laid by the World Turtle, and many similar stories can be found
throughout Polynesia. In Tibet, where egg myths abound, some of the
myths recount the origins of different orders of people. One myth, for
instance, tells how the water spirits from four original eggs produce
the classes of people who make up Tibet's social order. From the water
spirit of a golden egg came those who are kings, and those who are
servants came from a turquoise egg. An iron egg produces religious
leaders or holy men, and social outcasts come from a bronze egg. In
similar fashion, another myth tells how Tibet's six traditional clans
originate in six yellow eggs that are carefully cracked open by a
blacksmith sent by the spirits. According to this tale, two great
birds lay eighteen primal eggs that are yellow, blue, and white, the
colors of the earth, sea and sky. The clans of people emerge from the
six earth-colored eggs after a blacksmith from the gods cracks open
six white eggs and a smith from the water spirits opens six blue
eggs.
As with Pyusawhti's origin, there are also Central and South American myths in which semidivine heroes (often twins) are born from an egg (or a pair of eggs) which are found by a woman or by a married couple (typically peasants) who then raise the heroes hatched from the eggs. Sometimes the source of the eggs is implied to be divine providence but generally mysterious so that it is unclear if the eggs were ever even laid by anyone or rather perhaps created spontaneously. [But see also The Moon's Sister and Her Descent into the Orinoco below.]
A virtually universal phenomenon is a creation myth in which the first god or the first man (often these categories amount to the same thing) hatches from an egg which then becomes the universe; or that this deity fashions the world into an egg-shaped structure; or a combination of these two events occurs.
Sometimes the egg is laid by a gigantic bird, but this creature is also mysterious in the sense that, while it—or rather she—might essentially be the mother of a god, she is not necessarily considered to be divine herself, and this is the last occasion in which she features as a character in the mythical chronology.
A notable exception comes from a Greek myth in which Nemesis's mother Nyx—whom the Funk & Wagnalls Standard Dictionary of Folklore, Mythology and Legend interprets as the primordial cosmic bird—becomes the first mother after producing a god through an egg which she has laid. [Further details below under Primaeval Night: Grandmother of the Birds.]
In his book Creation Myths of the World, David Adams Leeming mentions that the Iban Dayak of Borneo have a story in which two immense birds, named Ara and Irik, "floated on the primordial waters and produced two giant eggs that became the heavens and Earth." Leeming also refers to these birds as "creator gods".
On the Greater Sunda Islands
Most variations of the creation myth of the Toba Batak of Sumatra Island start off with the chief deity, called Mulajadi Nabolon, dwelling in the uppermost level of the universe together with a few animals, including some birds. One of these creatures is a blue chicken with an iron beak and shiny, braceleted copper claws, named Manuk-Manuk Hulambujati.
This chicken laid three eggs which it at first complained to Mulajadi Nabolon were too big for it to incubate. After receiving assistance and specific instructions, however, the eggs eventually hatched, producing offspring who were, at the same time, the first men as well as the gods who organised the rest of the universe and who became the ancestors of the human race.
One of these divine men, who are also regarded as the sons of Mulajadi Nabolon, is named Batara Guru, who features in a story from the neighbouring island of Java, in which he and his two brothers were produced from a single egg laid by their mother, a goddess called Dewi Rekatawati, or Dewi Rakti, who was married to a god named Sang Hyang Tunggal. The Javanese Batara Guru, also called Sang Hyang Manikmaya, was the lastborn of his brothers, having emerged from the egg's yolk while the elder two came, respectively from the eggshell and the egg-white.
The Batara Guru from Sumatra had a daughter named Si Boru Deak Parujar, whose first child, according to Wikipedia, was egg-shaped, "not similar at all to humans". On the instructions of her grandfather Mulajadi Nabolon, Si Boru Deak Parujar and her husband buried this child and "out of it came plants that spread on the surface of the earth."
A Quintuple Deity, & Twin Gods, Among the Inca
According to the Huarochirí Manuscript, a storm-god named Pariacaca existed in the form of five men who were initially falcons which hatched from five different falcon eggs. Bizarrely, they are said, while they were yet still eggs, to have had a son, namely the hero Huatya Curi. The Manuscript also reports the idea that this Pariacaca is the son of the god Cuniraya Viracocha. It is not explained whether Cuniraya Viracocha may have mated with a falcon (or something?) to engender such progeny.
Mt Pariacaca, May 2004, by Cordillera Pariacaca
In the Handbook of Inca Mythology, Paul R. Steele & Catherine J. Allen write the following:
The Augustinians were told that a creator deity named Atajugu
lived in the sky. He divided himself into three persons with two
servants. He sent to earth a double, his servant called Guamansuri,
who disguised his divinity by working in the fields of the
Guachemines, the indigenous group in the area. Like the Andean deities
Viracocha and Cuniraya Viracocha, Guamansuri adopted the disguise of a
poor man. He then made a Guachemines woman called Cautaguan pregnant,
and was subsequently burned alive for this act. Cautaguan also died
while giving birth to two eggs. The eggs rested on a dunghill, and
eventually two brothers, Catequil and Piguerao, emerged. They revived
their mother and then began to kill the Guachemines. Those who escaped
were forced to flee to lower warmer lands toward the coast, and the
depopulated region was filled with a new generation of people. In
expelling the Guachemines, Piguerao and Catequil used the weapons
associated with the Andean Lightning deity, who created atmospheric
effects by firing sling stones.
The account also seems to imply that Cautaguan eventually became a mountain-goddess called Mamacatequil.
Unconventional Midwifery in Mongoland
A sort of reverse version of what you're asking about takes place in a story from the Democratic Republic of Congo. The Mongo hero Bokele was hatched from an egg. In Ntomba, one of the dialects of the Mongo language, according to African Mythology: A to Z, a book by Patricia Ann Lynch & Jeremy Roberts, the hero's name is Mokele, whose father, the chief of the Mongo people, was also "the first male human".
In another book, The Lianja Epic, by Mubima Maneniang', this early chieftain is called Waku-Waku, while, according to A Dictionary of African Mythology: The Mythmaker as Storyteller, by Harold Scheub, his name was Wai, and his many wives all become pregnant at the same time.
In his book African Myths of Origin, Stephen Belcher tells us that Bokele's father went off on a trip, "ordering that all his wives must have children upon his return". The favourite wife "showed no signs of pregnancy until she was helped by a forest spirit who drew the egg from her and hatched it." In Lynch & Roberts' account, while his father was out hunting,
Mokele was born from an egg that a sympathetic river goddess implanted
in his mother’s belly. The baby would not grow in the mother’s belly,
so the river goddess removed it and hatched it herself. The next day,
she presented the baby boy to the mother but took him home again for
the night. Mokele grew into the size of a four-year-old boy the next
day. The river goddess allowed him to go home with his mother but
warned her to wait a few more days until she showed him to anyone.
Within a few days the hero was full grown, had been dramatically introduced to his father, and was off on a mission to buy sunlight from a neighbouring rival community.
The Moon's Sister and Her Descent into the Orinoco
Watunna, the creation cycle of the So'to of Venezuela, tells us that, in the sky, the creator Wanadi kept a massive stone egg which was full of his unborn people, whom he wished to eventually live on the earth. According to Mineke Schipper, in Humanity's End as a New Beginning: World Disasters in Myths, "This egg, called Huehanna, was round like a ball, huge and hollow, its shell thick, hard and heavy" and the egg was abuzz with the laughter, singing and dancing of the people inside it.
Nuna, the moon, who was villainous and always hungry, stole Huehanna intending to devour it, but Nuna's young sister Frimene, finding the egg to be beautiful, and with a view to rescuing its contents, hid Huehanna in her vagina, telling herself that she would become the mother of the people within it.
A tussle in the night, in which Nuna unsuccessfully tried to pull Huehanna out of his sister, resulted in the first menstrual bleeding. Once Frimene confirmed that it was indeed Nuna who had accosted her under the cover of darkness, she fled from his house and into the jungle, running until she reached the newborn Uriñaku (Orinoco). Plunging herself into the river, she transformed into Huiio, the Great Snake, whereupon she built herself a house at the bottom of the rapids.
Frimene had been betrothed to Wanadi's brother Müdo, whom Wanadi commissioned to lure the Great Snake out of the water so that he could retrieve his egg from her. Müdo gathered the first-ever hunting party, which pursued Huiio along the river and shot her dead with arrows. Huehanna, which Huiio had released from her grasp as she was dying, burst open against a large rock in the water. The people inside turned into fish eggs, which, upon hatching, produced fish and the other animals which live in rivers and lakes.
Two of the fish-eggs remained intact. A jaguar named Manuwa, whose wife was a toad called Kawao, wished to keep these eggs to eat later, and instructed Kawao to take them home, although she rather wanted to hatch children to rear from them.
The twin brothers Shikiemona and Iureke emerged from the eggs and rapidly grew into unruly, childlike men, who not only raised a terrible ruckus, shouting, screaming and fighting, but also frequently changed themselves into different animals, such as fish, crickets and cockroaches, playing pranks on people.
After visiting Huiio's underwater house in the form of fish, Shikiemona and Iureke discovered their true origin when Huiio appeared to them in a dream. They avenged her death by flooding the entire earth. The rainbow, which was formed from Huiio's crown of feathers, remained as a memorial of her, and after the Great Water subsided, Wanadi decided to make other people.
Huiio herself, in actuality, was too powerful to die, and it is merely her body which remained on earth, while she went on to dwell "in the highest Heaven, in Lake Akuena", whose mistress she became, presiding over eternal life.
Mexican Maize Heroes and Founder-Kings
There is a Nahua myth in which an old childless couple finds a pair of large chicken eggs and they decide to eat one while giving the other to their hen to hatch. After a few days a boy emerges from the remaining egg, causing the couple to lament that they had probably eaten his sister who would have hatched from the first egg.
As narrated in their book A Land Without Gods: Process Theory, Maldevelopment and the Mexican Nahuas, Jacques M. Chevalier & Daniel Buckles conclude the story with: "Moreover, they didn't know who the little boy was." He was Sintiopiltsin, "the venerable son-god-ear-of-corn".
Kondoy, the king who founded the Mixe nation, was also hatched from an egg found by an old peasant couple who had no children. After having their turkey hatch the egg, they raised the boy that emerged therefrom as their own.
In his book Art and Myth of the Ancient Maya, Oswaldo Chinchilla Mazariegos reports that:
The father of Homshuk, the Popoluca maize hero, died in a distant
place—in war, according to a version from Soteapan, Veracruz—and his
mother ground the baby, tired of his ceaseless crying... Homshuk
somehow survived, and according to some versions took the shape of an
egg, which was found by an old cannibal woman. Her husband wanted to
eat the egg, but she decided to raise him.
Conflict then arose between Homshuk and his adoptive parents, the man still trying to eat him at first, and eventually the maize hero managed to kill the couple through trickery, continuing his adventures thereafter.
In a Yucatec story, an old witch, desirous of a child, prays to the god Chic Chan, who, according to Emese Fromm's article "The Pyramid of the Magician: Ancient Maya Ruins in Uxmal," was "one of the four rain serpents, or rain gods" who were representations of the god Chac.
One night she had a dream where this god told her to look for a
turtle egg by the nearby cenote, take it home and wait for it to
hatch. In the morning, the old woman did what the god told her and she
found a large, green turtle egg at the cenote.
She took it to her home, wrapped it in cotton and sat it in a
corner of her hut. She cared for it day after day, keeping it warm,
talking to it, singing to it. A few months later, the egg cracked and
a tiny baby boy stepped out.
The Yucatan Times article "Uxmal Yucatan: The Legend of the Dwarf" says that the witch named the boy Saiyawincoob and that "Time passed but the child remained the same size, the woman noticed that his voice changed and his beard began to grow, so she discovered that he was a dwarf." In Xbalanque's Marriage: A Commentary on the Q‘eqchi‘ Myth of Sun and Moon, Hyacinthus Edwinus Maria Braakhuis says that the dwarf eventually took on the name Ez. He is renowned for having magically built the famous pyramid of Uxmal in one day, thereafter becoming the city's king.
Pyramid of the Magician, 25 November 2009, by runt35 on Panoramio
Vietnam's Children of the Dragon and Grandchildren of the Fairy
Lạc Long Quân, the "Dragon Lord of Lạc," who had emerged from the sea, was the son of a mountain god and, according to Wikipedia, "the dragon goddess that rules the sky and the ocean." Lạc Long Quân married Âu Cơ, an immortal mountain fairy who often took the form of a bird.
After their union, Âu Cơ gave birth to an egg sac containing a hundred eggs, from which, seven days later, hatched their one hundred sons who went on to become the ancestors of the Vietnamese. Lạc Long Quân is quoted by Wikipedia as saying to his wife, after the hatching:
"I am a descendant of the Dragon, you are [a] descendant of the Fairy,
fire and water cannot live together in harmony." The two of them then
divided their children. Fifty sons followed their mother to the
mountainous north, the other fifty followed their father to live in
the south; these children are ancestor[s] of Bách Việt. The oldest
brother followed Âu Cơ to Phong Châu (Phú Thọ), became Quân's
successor and ruled as Hung King.
City of Thủ Dầu Một, uploaded to Wikimapia.org by thachhan120282, c. 2011
Korea's Hero Arisen from a Fallen Goddess
King Dongmyeong, who founded Goguryeo, the northernmost of the Three Kingdoms of Korea, was originally called Jumong. In one version of her adventures, his mother, the goddess Yuhwa, is disowned by her father and exiled into a mortal existence. Once she loses her divinity, upon being exposed to sunlight, she lays an egg, from which Jumong hatches. In his book Asian Mythologies, Yves Bonnefoy spells the hero's names as Tongmyǒng and Čumong, and his mother's name as Ryuhwa, narrating their story as follows on pp. 296-297.
The Wei chou (chap. C) relates that the eldest daughter of the
deity of the waters was bathing in a spot called The Heart of the Bear
when she was surprised by Hämosu, her future husband. The birth of
Čumong resulted from this encounter.
"In the capital of Eastern Fouyou, a man appeared, it is not
known from where, who said his name was Hämosu, and who claimed to be
the son of the Heavenly Emperor who had come to found a state." The
daughter of the deity of the waters explained how she came to know the
son of the god of the sky: "I saw the man. After he had lured me into
a house located at the spot called the Heart of the Bear ... he made
me commit adultery with him. After that, he left and never
returned."
In this way the girl conceived an egg from which Čumong emerged.
The same text says:
Ryuhwa... was shut up in a room. The light of the sun fell on
her. Shielding her body, she avoided it, but the light of the sun
pursued her. After that, she conceived and gave birth to an egg. The
king (of Fouyu) ... had the egg thrown to a dog, and then to a pig,
but neither the dog nor the pig ate it. It was abandoned on the road;
cattle and horses avoided it. Later, it was left in a field; the birds
covered it with their wings. The king ... tried to break it, but
without success. He gave it back to the woman (who "gave birth"). She
wrapped up the egg and put it in a warm place. A boy broke its shell
and emerged from it.
Tomb of King Tongmyong, from Persephone's Blog, on PersephoneTravels.com
A Dragon Princess and the Kingdoms of Burma
Pyusawhti (or Pyuminhti) was an early ruler of Pagan (now called Bagan), the Wikipedia article on him saying:
The pre-Hmannan Burmese chronicles claim that Pyusawhti, a
descendant of a solar spirit (နေမင်းသား) and a dragon princess
(နဂါးမင်းသမီး ဇံသီး), founded Pagan in 167/168 CE, and hence the
Burmese monarchy. The dragon princess, granddaughter of the Dragon
Emperor Kala Naga, was impregnated by the solar spirit who was
visiting the earth. Out of this union, the dragon princess laid three
eggs, all of which a hunter took away. The hunter accidentally broke
the gold-colored egg of the three at Mogok, and the broken golden egg
turned into numerous rubies and gems (for which the Mogok region is
known to the present-day). The hunter then lost the remaining two eggs
during one heavy storm. One egg, in brown color, ended up in a small
kingdom in either northern Burma (Thindwe or Tagaung) or Yunnan, and
out came a female human princess, who later became queen of that
kingdom. The remaining egg, in white color, drifted down the Irrawaddy
all the way to Nyaung-U, where it was picked up by an elderly
childless Pyu peasant couple. When the egg hatched, Pyusawhti was
born. The Pyu couple raised him like their own son. He was then
educated by a local monk named Yathekyaung.
King Pyusawhti, 26 October 2015, by Phyo WP
Heroes of the Ovambo
The Ovambo of Angola and Namibia tell of "a good girl who married in a faraway country" in what Harold Scheub seems to interpret as the beginning of time. She laid an egg several times but before long each one broke, until her mother advised her to place the next egg that she laid on top of the granary, which the girl did. When the egg hatched, a child emerged from it who named himself Nambaisita, the "Self-Created" One, boasting that he had made himself and that he therefore did not fall under the jurisdiction of the ruler of the world.
Kalunga (God), who lived in an enclosure in a different neighbourhood, was annoyed by this threat to his authority, and summoned the boy to his home, where the two contended with each other in a series of challenges set by Kalunga, all of which Nambaisita overcame. According to Edwin Meyer Loeb's Kuanyama Ambo Folklore, Nambaisita even went so far as to overthrow Kalunga and become God. Scheub spells the hero's name as Nambalisita, regarding whom Christian Rodrigues Fischgold mentions, despite his boasts of self-creation, that there is at least one version of the story in which Kalunga is actually the boy's father, which relationship might be at the root of their rivalry.
In another story, a girl called Nehova, "who belonged to a tribe of people whose women laid eggs", was abducted by a king who killed her brother and married her. When she laid eggs, the king broke them, thinking they were being laid in her hut by a hen. Nehova then concealed the next egg she laid on top of her corn granary. Many days later, when the king left with his household and all his wives for a feast in a different homestead, she stayed behind, knowing that the hatching time was at hand. The boy who hatched from the egg immediately went on to resurrect his uncle. When the king returned home, his own newborn (or rather newly-hatched) son killed him, and the deceased ruler became an evil spirit.
Tibetan Heroes, Demons & Deities
The Epic of Gesar Khan is of great importance in Tibet but also exists in Mongolian, Buryat, Balti and Ladakhi versions. In one of the numerous variants of the story, Gesar is the son of a divinity named As Kenzo, sired upon a Nāga princess called Dzeden. The Nāgas are a race of subterranean underwater snake-people, making Dzeden something like a serpentine mermaid.
Her son Gesar is also the incarnation of a bodhisattva named Thubpa Gawa. She becomes pregnant with him when As Kenzo one day appears to her and gives her a drink of holy water impressed with the reflection of Thubpa Gawa's face. A short while thereafter, as narrated in The Superhuman Life of Gesar of Ling, by Alexandra David-Néel & Lama Yongden:
[O]ut of a white vein, which opened on the top of her head, came a
white egg marked with three spots that resembled three eyes...
A little while later it broke of itself, and from it emerged a male
child with dark skin, the colour of beer. He had three eyes.
Dzeden is too tripped out by the strangeness of how the child and its egg had come about, and she uses her thumb to put one of the baby's eyes out. This child, who was already talking before his egg came out of his mother's head, then goes on to have adventures shortly thereafter, eventually becoming a great conqueror of the surrounding regions. Like the Korean Yuhwa, Dzeden also lives a human life. Her people the Nāgas are also shape-shifters, and thus she dwells in disguise as a peasant girl named Gongmo, in which form she raises her son.
Mandalas.Life Tibetan Statue of Naga Kanya
In another story, the violence demon Tsi Mar, or Tsiu Marpo, hatches from a red egg of blood, laid by a flesh-eating demoness called Dongmarma, the consort of the savage demon lord Lekpa.
There is also a Buddhist story about a pair of deities, brother and sister, whose parents, a man named Nujin Zangki Ralpachen and his wife Sinmo Dragi Ralpachen, had produced them from a pair of eggs. For the whole story, see the Himalayan Art Resources article "Begtse Chen (Buddhist Protector)," particularly the section The History of the Protector Deity Begtse.
And Back Around to Ancient Greece
Molione and Her Twin Sons
As aforementioned, there is a very close Greek parallel to the story of the union of Leda and Zeus. Leda's cousin Molus had a daughter named Molione, who in turn had twin sons, Eurytus and Cteatus, who, from their mother, are known as the Molionids. From Molione's husband Actor, these brothers were also called Actorides although it was claimed that their real father was the sea-god Poseidon.
According to Athenaeus' Deipnosophistae, the Molionids were hatched from a silver egg. According to Theoi.com, their mother "was seduced by the god Poseidon in the guise of a bird." I have not found any ancient source which explicitly says that, so perhaps it is merely that website's interpretation, albeit not an unreasonable one. Poseidon does undoubtedly mate with Medusa in the form of a bird in Ovid's Metamorphoses, which is supposed to explain how their son, the winged horse Pegasus, comes to have the form that he does. Pegasus's birth itself is fairly unconventional, but no mention of an egg appears in accounts thereof.
Presumably Molione is the one who laid this silver egg, and Poseidon becoming a bird is supposed to explain how an apparently regular, garden-variety mortal like her ends up laying an egg instead of giving birth. The closest that Molione seems to have been in terms of possessing biological divinity is that her father Molus was a son of the war-god Ares.
Thyestes' Sister
Among Helen's in-laws, her husband Menelaus had an uncle named Thyestes. The Dictionary of Greek and Roman Biography and Mythology cites the Lexicon of Hesychius of Alexandria, and Tzetzes' commentary on Lycophron's Alexandra, as the sources for an obscure story in which Thyestes has a sister named Daeta (or Daeto) by whom he becomes the father of a certain Enorches.
For some unexplained reason, Enorches was hatched from an egg. As with Molione, there does not seem to be anything special about Daeta that would cause her to give rise to offspring by egg-laying. Assuming both of her parents were the same as those of Thyestes, then her closest connection to the gods would be her great-grandfathers Zeus, Ares and Atlas, a pedigree she would share with numerous other ordinary mortals.
Primaeval Night: Grandmother of the Birds
In Aristophanes' play the Ornithes ("Birds"), the flying creatures therein, being the title characters of this production, boast of how much more ancient they are even than the Olympian deities. A Chorus of them narrates their origin, telling us that "At the beginning," before the Earth, the Air and the Sky existed, "there was only Chaos [the Void], Nyx [Night], dark Erebus [Mists of Darkness], and spacious Tartarus [the Abyss]."
Firstly, black-winged Nyx laid a wind-egg in the bosom of the unending
depths of Erebus, and from this, after Seasons rolled around,
sprang the graceful Eros, on his back the glitter of his golden wings, swirling like
the whirlwinds of the tempest. In broad Tartarus, he mated with dark
Chaos, winged like himself, and thus hatched forth our race, which was
the first to glimpse the light.
A striking parallel to this occurs in Hawaiian mythology. In the poetic chant called the Kumulipo, one of several couples born of the primordial darkness are named Po’el’ele, the personification of the darkness of night, and Pohaha, the personification of the darkness just before dawn. These become the parents of a few different kinds of insects and they also produce an egg from which the first birds hatch.
In an Egyptian creation myth, the sun-god Re first appears as "the bird of light," dispelling the primaeval darkness in the form of the first sunrise after hatching from an egg laid by a celestial goose, or otherwise by an ibis.
Ptolemaic Egyptian Goose, Walters Art Museum
