As they say in Hawai'i, mo butta is mo betta, brudda!
I always use butter to fry eggs, any style, and as Bobby Flay says "anyone who doesn't cook eggs in a non-stick pan is a fool."
Next time make an Italian omellete instead of French. It's called a frittata. There are plenty of recipes on the internet, and there are many variations, most of them wrong, believe me. I know some will object to my rules, but I am Sicilian (pronounced Sig-a lee-on). I know these things!
Here are some basic frittata rules:
A cast iron pan is best.
5 or 6 eggs for an 8-nch pan, 8 eggs for a 10-inch pan, up to a dozen for a 12-inch pan.
If the recipe calls for milk, it's a souffle, not a frittata.
If the recipe calls for flour, it's a queche, not a frittata.
Small diced potatoes are traditional, but not absolutely mandatory. I like them.
It needs some kind of meat. I like Italian sausage, diced ham, or carnitas, but if you are keeping kosher, chopped up steak, chicken, turkey, ground beef, shredded leftover leg of lamb, goat, etc. work well.
Peppers are nice. I use diced bell peppers (any and all colours), chopped jalapenos, and/or sliced pepperoncini.
You cook it on the stove top, moving the egg around until it's cooked through, then you add a melting cheese like mozzerella to the top and stick it under the broiler until the cheese is melted and slightly browned.
You can slide it out onto a plate, but I usually slice it like a deep dish pizza and serve it right out of the cst iron pan.