Wouldn't the creature hatched from this mutated egg only really become a "chicken" by successfully reproducing, and passing its "chickenness" to a new generation of creatures? Beyond that, wouldn't there need to be enough successful generations of "chicken" reproduction in order to cement "chickens" into the biosphere, for long enough that humans evolve and identify them as "chickens"?

Only after that could you look into the past and say that egg was the first "chicken egg." At the time the egg was laid, and I would argue when the egg mutated, and then also at any time before "chickenness" was defined, there would be no way to correctly assert that a specific egg was "chickeny".

So ... "chickenness" must have come into existence first, and only then could it be possible to look back in time and identify the first egg that qualifies as a "chicken."