I'm late to this discussion, but you might want to invest in a separate pan for eggs and fish. Here are two Wirecutter pages on pans -- one non-stick (with lengthy discussion on non-stick surfaces) and one on the best pan (which is NOT non-stick, but also not what you want). Archived links so you don't have to worry about paywalls, I hope : Best Skillet | Best Non-stick

tl;dr: Use an absolutely flat bottom with flared sides and good balance. 10-inches is perfect for omelettes, but use a bigger pan for multi-person meals. Beyond that, "Nonstick pans are best for cooking things like eggs or delicate fish fillets, but they’re not appropriate for high-heat jobs. The slick surface also can’t develop the fond (tasty brown bits that develop on the bottom of a pan) that’s integral for sauces."