Seems like the question of whether the chicken or the egg came first is becoming moot. A hen in Sri Lanka gave birth to a baby chick, without an egg. The event was confirmed by veterinarians, although it's not as mysterious as you might think.
The mom actually died soon after giving birth. The autopsy found that the hen had developed an egg, but just never expelled it. It incubated for 21 days, and the chick hatched inside the egg. The internal injuries this caused was what killed the mother.