Filling: I use whatever is available at the moment. This time, for one two-egg omelet, I use red onion (thinly sliced, 1/3 medium), fresh shiitake mushroom (stem removed and sliced, 4-5 medium), leftover oven roasted pork fillet (4-5 thin slices cut into strips), baby spinach (a very small handful) and fresh goat cheese (3 tbs). I first saute the onion in the mixture of olive oil and butter, add the pork and shiitake mushroom. After they are cooked, I add spinach and saute until wilted. Season with salt and pepper. Since nice "fond" is formed, I deglaze it with a Marsala wine (1 tbs, optional) and cook it until liquid is all gone. I add crumbled goat cheese (I do not mix at this point) and leave the mixture on the stove on a very low flame (our stove has a very low simmer setting or use as a "warm" setting if you are using an electric stove top) (#1, image below).
Eggs: I used two brown eggs (for one omelet) beaten with the addition of cream (2 tbs) and seasoned with salt and pepper. If you use 3 eggs, it will be easier to form an omelet but you can do it with two eggs.
In a non-stick frying pan with low side or an omelet pan (8 inch) on a medium low flame, I melt butter (1 tbs) and pour in the egg mixture. When the bottom is set, I use a small spatula to lift one side of the omelet and let the excess uncooked egg flow under the cooked portion. I repeat this on all four quadrants until the desired amount of liquid egg remains (I like it almost all cooked). If you want to make pristine yellow, not-browned omelets, you have to cook the eggs on a very low heat with a lid on to cook the omelet through (for a stuffed kind not for a plain French style) but I like some brown markings on my omelets. I add the filling, which was kept warm allowing the cheese to melt, onto half of the omelet--the half opposite the handle of the pan (#2 in the image below). Take a plate in the left hand and hold the frying pan in the right hand grabbing the handle from underneath (I am right-handed), angle the pan and slide it out and then fold over the omelet using the edge of the frying pan to cover the stuffing (I should have recruited my wife to take pictures). The omelet is done (#3 in the image below). I am not sure what this shot (#4) is called in food bloggers parlance (I have to defer to Jon), here is the cut section.