Based on the jpeg, this meme came first

I recall reading somewhere that it would have been a proto- chicken kind of thing. Like not quite a chicken but it laid an egg and the first chicken came out.

Maybe a gene mutation of some sort.

Thats on the branch labeled traitors that leads to paying bills.

Yeah, this is silly (and fun) but avoids the real problem of course. The question can be like you said, "which came first, the chicken or the chicken's egg?" And for those that still want a literal answer, wikipedia says:

If the question refers to chicken eggs specifically, the answer is still the egg, but the explanation is more complicated.[8] The process by which the chicken arose through the interbreeding and domestication of multiple species of wild jungle fowl is poorly understood, and the point at which this evolving organism became a chicken is a somewhat arbitrary distinction. Whatever criteria one chooses, an animal nearly identical to the modern chicken (i.e., a proto-chicken) laid a fertilized egg that had DNA making it a modern chicken due to mutations in the mother's ovum, the father's sperm, or the fertilised zygote.

As an alternative, though it's a bit more of an ungainly mouthful, I like: "which came first, the first species to lay an egg or the egg of the first species to lay an egg?" That one is a bit harder but you might still be able to tease out an answer. That way I think it gets a bit more into the problem of qualitative vs quantitative when you do (which is partly why I say below that this is related to the problem of the heap). Of course it's really meant to be a philosophical problem anyway, and in that sense, it remains a paradox. It's a way of making an analogy for a "causation dilemma" and gets at the idea of infinite regress and the paradoxes that brings up. It's also related to the sorites paradox or the problem of the heap, which actually is an element discussed in Marxist (more because of Engels) dialectics.