As a small child my father would often ask me this question and I would go away and mull over the question for ages ---come to think of it , I still do not know the answer in spite of my many theories !!
Modern day birds, scientists agree, are closely related to two-legged meat eating theropods like T-rexes and velociraptors. So i'd guess the egg came first. Good shot.
@gamelee This is closest to my theory as well -- although not very knowledgeable on named dinosaurs ,- the fact some laid eggs > flew. >evolution.> birds !!