In terms of evolutionary biology, the egg would have to come first.
Consider this: Say the direct ancestor of the chicken (the thing that gave birth to the first thing we'd call a chicken) was a pterodactyl (again, hypothetical, and i just like that word). When that thing laid the egg, the egg had the mutations to make it a chicken. The mutations happen before the egg, not at hatching - thus the only logical explanation is that the egg, technically, came before the first of whatever organism hatched from it