I too whisk the eggs together without seperating. I like to add a bit of milk to the mix.
My prefered cooking method is to pour the egg mixture onto my large electric griddle in a rectangular shape, cook over low heat, with butter until the egg begins to firm, then add the fillings, which always includes some kind of cheese, and if I have them handy, some saute'd mushrooms. Then the other topping that Ihave on hand go on. I fold one side over the middle, and then the other to make an open-ended burrito shape.
When making omelts for my wife, I make them in a round cast-iron pan over low heat, and immediatley after adding the egg, push the solids portion to the center. This causes the edges to firm up quicker. I then slide the cooked side onto a plate and flip the whole thing back into the pan. I place toppings on one side, and then, as I'm sliding the omelet onto the plate, I fold the other side over the top. She won't have them any other way.
Favorite toppings include freshly sliced tomato, Velveeta, American, or Muenster cheese, saute'd mushrooms, Diced onion that have been sweated until sweet, S & P, cooked and crumbled breakfast sausage, or diced ham, diced green peppers (again sweated like the onions), chopped black ollives. Sometimes I'll add garlic powder to the egg, or maybe some choppped oregano.
If I'm looking for a more Tex-Mex flavor, then I'll add some fresh cilantro and garlic to the raw egg, and use cheddar cheese, or a good Mexican cheesem along with some Mexican style chorizo with some spicey chopped pepper thrown in.
For me, omelets, when done right, should be tender and moist, but should not be mushy. And that is a bit of a trick. Timing and temperature are everything.
Seeeeeya; Goodweed of the North