The egg came first. Dinosaurs are supposed to have evolved into birds. Dinosaurs were laying eggs millions of years before birds existed. the first bird was probably born from the egg of some sort of evolutionary link between birds and dinosaurs. Based on this theory The first chicken would've come from an egg.
Only way that would not be case is if scientists are wrong and chickens evolved from some sort of mammal that had live births like puppies. and later switched to laying eggs afterward somehow.