An omelette is a standard test of a chef, like a Dry Martini is a standard test of a cocktail barman.
So this is how I was taught. (I had to do them in front of guests at the table along with other stuff like "Crepe suzette".)
We used copper frying pans which heat up quickly like feck!! and cook incredibly evenly.
Heat the pan up really hot, stick butter in it and roll it all over the base and sides, chuck the egg, salt and pepper in, (some add a tablespoonful of water for each egg, it makes it more "mobile") and slosh it around including using the sides, lifting the edges to let the liquid flow to the pan to cook. (You should be able to do all this in the heat of the pan).
Then add your filling, cheese, cooked mushrooms, ham, whatever. Cook it for a bit until you see the cheese melt and the egg kind of glazes, flip one half over on top of the other half, keep cooking it until it is done to your liking (Some frogs like a bit of raw egg to still be in the middle! Yuk!) then slide it out and eat it. should be slightly brown on the outside yet still soft on the inside.
LARVELY.
In the old days no one washed a cast iron frying pan, you re-seasoned it all the time. Pain in the bum but cooked better than more modern ones especially steel or ally ones where the base is thicker than the sides so stuff burns on the sides. Bastid trying to cook a decent omelette in one.
We have both steel and non-stick. chuck em all int dishwasher. Good quality ones can take it but we only use wooden or plastic utensils on the none stick ones.
I'll have to try your way with the cheese in the mix from the start. The cheese WILL give off fat which will to an extent stop it sticking.
Look at us, regular WI meeting!!!
Raw in egg in the middle err no thank you
I used to make them like you said but one day I justlobbed everything in the bowl, I usually put a some topping on then fold it then its a top filling
We have a well burnt pan as well
Good if you want black n yellow scrambled egg. 
