Eggs sticking to the pan

If it's a stainless steel pan then the eggs should not be sticking. Your pan is too hot I think. It is always best to cook eggs over low heat, they won’t stick that way. I always remember this: “More butter less heat”.

If the butter or oil burns, then lower the heat. Remember that the longer the pan stays on the heat, the warmer it will get; maybe when you cooked your first egg, the pan is not fully heated.

I would clean (wipe clean) the pan between each batch of eggs to remove the oil and butter residue

If the oil/butter is burning, it will impart a burnt flavor to the eggs. If you cook on low heat, all you will taste is the butter and eggs. So based on what you wrote, try turning the heat down. That the oil/butter is burning is a red flag...for me anyway.

I don't like non-stick pans but I use them for eggs. I put a pat of butter in the pan and swoosh it around to coat the entire pan. I add my eggs but I am using very low heat the whole time and they never stick, at all.

The majority of my other pans are stainless steel. I don't have any cast iron. I don't remember stainless steel sticking but I'll try them the next time I cook eggs.

When I cook eggs, I basically melt the butter and coat the bottom. I add the eggs before the butter can ever bubble. At the low heat I'm using, it's unlikely to anyway. I usually make scrambled eggs but fried are the same.

Sounds like you're using too much heat.