The Grilled Cheese

This question does not pose an impossible problem. Nor should the notion be used to describe other problems of causation. The puzzle can be solved with the help of a rhyme: Evolution provides the solution. That is, once we define a couple important terms. As long as the chicken in the question refers to the genetically modern chicken--or, indeed, to any specific past genetic version of chicken--and the egg to that object consisting of a calciferous shell that protecting the embryo of that version, we can continue.

Life evolves by genetic change. Genetic change occurs by mutation and by crossover. And in chickens, like in humans and most other forms of life outside of bacteria, these processes occur during sexual conception, when the male and female gametes combine to form an embryo. The first modern chicken embryo was formed this way, from two parents who weren't quite yet modern chickens. And this embryo developed a calcium carbonate shell, from which eventually hatched the first genetically modern chicken.