It's not even that simple. Evolution occurs gradually, so gradually in fact that there is no "first" chicken. Any given chicken/ancestor you pick will be so genetically close to its parents/children, that they're the same species by any definition.
Another way of putting the paradox is "Which chicken arrived first, the chicken or its parent"?
Chicken. If we establish some trait of a chicken we could be able to trace the first bird wich had the specific gene combination and it's parents had not. Still a well defined non paradox.
Another way of putting the paradox is "Which chicken arrived first, the chicken or its parent"?