When they say egg they mean chicken egg, not any kind of egg. It's not completely clear if the first chicken came up from what you'd call your usual chicken egg.

And who laid that egg? Certainly not me. The implication is that it's about a chicken egg and it's not difficult, it just doesn't have a determinative answer because chicken is a spectrum. What even is the first chicken? There ain't just a thing, that's not how evolution works. It's a gradual change from once species to another like language change. Who was the first speaker of modern English and how could they understand their middle English speaking parents?

That's why the chicken and egg thing is used as an allergy for questions that do not have a straightforward answer. Who started the fight? "Certainly not me, I made a joke and you took it seriously and then you insulted me" "I didn't hit you hard but you did"

To expand on this: mutations between generations happen exactly there, between generations. So the parents of the "first" chicken (if you draw the line somewhere on the evolutionary scale) were not chicken; the egg however was a chicken egg, as it contained a chicken.