not to dig this hole any deeper, but the defining characteristics of a chicken aren't like, easily identifiable. we can build a hypothetical in which two proto-chickens are genetically capable of producing offspring that is "chicken", but that's kinda rube-goldbergesque, there must have been some extremely specific series of genetic coincidences required to produce something chicken enough to be a "chicken" in that scenario. genetics, and evolution more generally, tends to be more complex. the specific genetic markers that distinguish chicken from non-chicken, if we say they exist, are probably not in and of themselves what makes a chicken, because single gene changes don't usually make creatures incapable of interbreeding with their parents' species, and that's a defining feature of the taxonomic category "chicken" belongs in.

like, if we grant that the chicken came from a proto-chicken egg, because the chicken has a special chicken gene, its really really likely that the next generation of "chickens" came from our progenitor chicken mating with a proto-chicken. taxonomically, that means that proto-chickens are chickens, because species is commonly defined by the ability to produce fertile offspring (eggs). so for every step in the process towards chicken-ness, we can't really say that the egg came first in a taxonomical sense, because the first member of the species of "chicken" (as defined by whatever genetic marker we claim indicates chicken-ness) was almost certainly able to reproduce with things that didn't have that genetic marker!

maybe there's some other sense in which the chicken and the egg can be discretely separated, but if we are talking about species, taxonomically, anything that can lay eggs to make fertile chickens must be a chicken by definition, barring some really weird edge cases that probably didn't happen.

fun fact: plants can do the weird edge case, and do it quite often. plants can duplicate their chromosomes without catastrophic consequences, unlike animals, and they can reproduce without sex with another individual, so a plant can produce offspring that aren't fertile with their parent species, and can reproduce independently (called polyploidy). so a seed can come before the grass (as with some kinds of wheat, and many other plants). this can also happen in reverse, where a polyploidal offspring can start reproducing with a species it couldn't before!