Depends upon whether I've already broken both of them into the pan. Growing up on the farm having free range chickens, the eggs would sometimes have blood in them for some reason. Mom always broke eggs into a saucer and poured it into the pan or mixing bowl before breaking the next one.
So, if you have a neighbor you really have a problem with and are of the get even persuasion, go pour it into the air intake on the front cowl of the vehicle. If you like animals, put it out at the far back end of the yard and let the wildlife have it. Or you could flush it down the commode, without the egg shell, of course. It's less likely to smell than if flushed down the sink.