This age-old question really has a simple answer. Attempts to answer it, however, and attempts to get around implications of the simple answer are often quite convoluted.
According to the Creator of chickens, and the author of the Record of their origins, chickens came first. It was on the Fifth Day of Creation Week that He created "every winged fowl after [their] kind" (Genesis 1:21) complete with the DNA to reproduce that kind. Then He "blessed them, saying, Be fruitful, and multiply" (v.22) using that DNA. For the chickens this meant lay chicken eggs. Problem solved.