There is such a thing as getting too technical and loosing sight of the goal of the question.
Which is exactly what you did in the opening post.
I think the question is intended to be paradoxical and is asking how can you have an egg without a chicken to lay it and how can you have a chicken without and egg to hatch from. When you start talking about evolution and completing a set of DNA you've already gotten too technical.
This is why, when confronted with the question, I ask for some clarification in the definion of the word egg(I know what they mean by chicken) because it can be ambiguous. Is it, which came first:
the chicken or the chicken's egg?
or
the chicken or the egg's chicken?
The answer to both of these questions is obvious. Leave out the clarification and the question becomes ambiguous, as it is trying to be a paradox.