Neither. And both. This is a false dichotomy.
You see, there is no such thing as a chicken, nor is there such thing as a chicken egg.
Let me explain.
Chickens aren't a static thing. They evolve over time. They are constantly changing.
So are their eggs and the contents of those eggs, too.
Many millions of years ago, there was a dinosaur. It looked vaguely chicken-like, but it had teeth and claws on its "wings". If you saw one at night, you might briefly mistake it for a chicken.
Over time, though, this creature changed. It's teeth disappeared, as did the claws on its wings. It gained the ability to fly, and then lost it again.
At what point did it become a chicken? It still isn't a chicken, remember? There is no such thing as a chicken.
The eggs you buy at the store come from a small dinosaur that is still in the process of becoming what it will eventually become. It is the first of its kind. It is the last of its kind. Its children will not be chickens, any more than it is.
They are beasts on the spectrum, categorized by we humans as "chicken" but no more chicken than dinosaur or incredibly advance amphibian or hyperevolved protozoa.
There was no first chicken. There was no first egg. We could arbitrarily draw the line anywhere we wanted to, summarily deeming that this is the first chicken or the first egg.
But, in the end, that decision would be arbitrary. The creature we are about to eat is what it is, regardless of where we draw that line. And, this is important, the being just on the other side of that line, regardless of where we draw it, is more "chicken" than any other creature. If you were to see one, you'd say, "that's a chicken". Either that, or we've drawn the line so far from what we think of as "chicken" that the creatures on both sides would fail this simple test.
False dichotomies of this kind are common. Which is better, conservative or liberal? Answer: yes. Which is worse, rape or murder? Answer: no.
Life is far more complex than "yes" or "no" and "better" and "worse". Expand your mind and see the world as it is: a spectrum of existence.