Which Came First: the Chicken or the Egg?

This is a factual, rather than a philosophical question. However, it is a legitimate task for philosophy to analyze the conditions under which it would be correct to say that the chicken came first, as well as the conditions under which it would be correct to say that the egg came first.
If the theory of Creationism is true, then God could have created the first chicken, which hatched the first egg, or He could have created the first egg, from which the first chicken hatched. Either task would have been equally easy (or difficult). Unfortunately, the information which would enable us to answer this question is missing from the Book of Genesis.

If Darwin's theory of evolution is true, then we can say that the 'trick' of producing a soup of proteins and fats enclosed in a hard casing, inside which an embryo is protected and nourished, was developed by the prehistoric creatures from which chickens evolved. We know that dinosaurs laid eggs. Dinosaurs are reptiles. The accepted view is that birds evolved from reptiles. So in that sense it would be true to say that the egg came before the chicken.

But what about that first chicken? What kind of egg did it hatch from?

If we had the power to go back in time to follow every line back of each one of the millions of generations that led up to the chicken that supplied your breakfast egg this morning, it would be impossible to identify the first chicken. There is no single characteristic, so far as I understand it, which separates a real chicken from a bird which is ever so much like a chicken, but is not a real chicken. However, supposing there is some unique, new feature, a crucial genetic mutation which separates chickens from non-chickens, it logically follows that the first bird to possess that new feature was hatched from an egg which was laid by a bird which did not possess that feature.