To my mind I think we can QED this eternal question.
Let's see... the answer is, egg first.
Seriously.
First, what are we trying to look for? we have to define them clearly; we can't find what we don't know we're looking for.
The egg in question is a chicken egg. Not a duck or goose egg. An egg that'd hatch into an animal that we call a chicken.
But what's a chicken? We didn't have this tool less than a century ago, but now we can define a chicken as an animal with a specific set of DNA that we classify as "chicken". Again, not duck gene, not goose gene -- chicken gene.
Okay, what's a chicken gene? Philosophers in milleniums has been trying to define what a chicken is by arguing what combination of features we'd consider an animal "chicken"; we may also argue to the end of time what particular sequence of DNA we'd identify this a chicken gene.
However, we don't need to do it. The only thing we need to define is, there is a certain DNA sequence we consider as chicken gene, even if we haven't select one as a definite chicken gene sequence. We just have to know it exists.
Additionally, we have to assume (and bare with this fact) that we will never be able to tell if mid-life gene mutation could have created the chicken gene, as unless we have a time machine and we managed to put every single creatures of all the near-chicken species over history (at least to the range we estimate when chickens started to exist), we can never catch the chicken gene mutated to existence.
Given all the definitions and assumption above, we can easily say that, for all our understanding of the life cycle of the birds (Aves), that their life starts as eggs, which contains the embryo that'd contain the same gene the animal that'd eventually hatch from, the undefined but existing chicken gene exists first in the egg that'd hatch into a chicken.
Or simply put, without being able to tell when a living non-chicken mutated into a chicken, the first chicken must have its chicken gene since the very first stage of its life, meaning an egg.
One last argument: some would say before the world has a chicken that egg is not an chicken egg as the concept of chicken doesn't exist yet. However, the concept of stars didn't exist until humans defined it yet the stars has been around since almost forever. The concept and form and features of X existed already, it's just we humans substitute the X with "chicken" on it.
