This has to be one of the oldest arguments ever. Which came first. There are two ways to look at it so I'll consider both.
Biblically, the chicken had to have come first. God made adult animals, the same way he made man the first man as an adult. I cant even imagine him dropping off baby animals in the garden of Eden. So looking at it that way, I guess the chicken would have been made fully grown and would have then laid the first egg.
From the evolutionary side of things, we have a different thought on this issue, The egg would have definitely come first. All aves are the result of evolution from reptiles. Theropods which were a prehistoric reptile evolved into the birds we have today. The theropods laid eggs that looked less like them for generations and eventually, more like birds. Then the Archaeopteryx was born. It was the first bird. It had teeth and vaguely resembles its reptilian ancestors but was technically still a bird. So therefore, the egg came first according to evolution