A  chicken is defined as a fully matured bird of the subspecies gallus
 gallus domesticus.

 A chicken egg is defined as an egg which  will  eventually  hatch  and
 will  result  in  a  bird that will grow up to be a chicken as defined
 above.

 It then follows that the last pre-chicken bird or proto-chicken on one
 sunny  day laid an egg with a mutation that caused this egg to produce
 a chicken rather than a proto-chicken.

 Hence, the egg came first.

 So there's that.

 But wait, that's just one egg! Where do all the  other  chickens  come
 from?

 The mystery continues.