The transition from a chicken to an egg from the reproductive process and laying allows for a changing genetic code, same how a child is different from the parent in subtle ways. If we assume evolution is the process we are abiding by then the creation of the species we call chicken may have been a transition from a very chicken-like species laying an egg which hatched into what we call a chicken.
However, the change from egg to newborn hatchling has no process that rewrites or even changes the genetic code, so its impossible for an egg to hatch an animal different than the DNA that originally took part in creating the egg specified (barring factors like toxins or radiation, but that mostly factor into birth defects and premature death rather than evolution).
Thus, it is possible for a non-chicken to lay an egg using material that would form a chicken (assuming they are very very similar species) but it is impossible to have an egg hatch into a creature different than dictated by its DNA. So the only changes in the evolutionary process come from the egg being laid, not the egg hatching. Thus the egg came first, as the transition from chicken-like bird to chicken has to come from a reproductive mutation, something that can only come from the laying of an egg.