I don't think this qualifies as a funny answer to the problem. I always thought of it in the way of evolution.
At one point way back the pre-chicken-like things ( so pre-dinosaurs-like things since we are told that's where birds came from) must have had live births. Over time since that world was a tad more fierce than ours, perhaps they starting birthing earlier still in the embryonic sac. Maybe for protection, maybe for fun - who knows. Then over time the sac evolved with the proteins mentioned and it became a hard encasement. In time becoming the egg we know today.
So technically then - the chicken-like-dinosaur-like thing came first.
mmmm dino-chicken, the other white meat
I thought that was pork lol