They really aren’t that hard, but lots of folks who claim to be cooks fear to make omelets…shame that, because once you learn they are great for turning whatever is in the 'fridge or pantry into a meal. I think they are excellent for beginning cooks, as the encourage experimentation, and drive home the point that you can screw up pretty badly and still put decent eats on the table.
You need few items of “special” equipment: A wire whisk (though I have used a fork in a pinch) and a small mixing bowl with tall sides. A smallish frying pan, preferably non-stick, but well seasoned cast iron is fine. Too large of a pan will be a problem, unless you have a large appetite, and make 4-5 egg omelets. I’ve never used one of those hinged omelet pans, and frankly can’t see the point. I’ve made omelets backpacking with dried eggs, in a mess-kit frying pan, so don’t let lack of equipment stop you.
~1 teaspoon water per egg. Not critical, less for fresh eggs, more for ones that have been in the fridge for a couple weeks (they lose water over time). Measure till you get the hang of it, but eventually you’ll just splash some in, based on how the eggs came out of the shells.
Salt and pepper to taste.
Whisk the eggs and water to a foamy froth. Pour into a hot frying pan (see WhyNot’s advice above, but I might suggest olive or canola oil instead of butter)
Now for the tricks:
The omelet will tend to dry out on the edges, which are thin in normal shaped frying pans. To prevent this, use a spatula to smoosh the thin edges toward the middle. At the same time, tip the pan to allow runny egg mixture from the middle to run under and around the smooshed in edges. No problem if it ends up a little thicker around the edge. Do this two or three times around before folding.
Wait until there is only a little runny egg on top. You can fold in half, or in thirds. Takes practice. Don’t worry, you can totally mess it up, and still have a yummy meal. (And you will mess it up the first 10 times at least!) Practice so you can make it come out pretty for company though. Don’t wait until the top is dry, or the “hinges” will break.
The omelet can continue to cook on the bottom after folding, but to finish the cooking, you need to turn the whole shebang over. With a lot of practice, loosen with a spatula, and flip flapjack style. Lacking such practice, lay a plate face down on top of omelet, turn plate and pan over, leaving omelet on plate, slide omelet off plate back into pan.
Cover pan with vented, or poorly fitting lid. A piece of foil will work fine. If you under-cook, you will have a little raw egg in the middle…doesn’t hurt a damn thing, and much less gross than burned egg. With practice, you will learn to get a nice lacy light brown on the outside.
Fillings, though, are what make omelets truly wonderful. I divide fillings into two classes, stirred-in, and folded-in. You can use one or both kinds in the same omelet.
Stirred-in. Mix these into the egg mixture prior to pouring into pan:
Chopped spinach, either fresh or frozen.
Chopped onion, either fresh or dried.
Crumbled fried bacon, or chopped ham.
Sprouts, bean sprouts or alfalfa sprouts.
Chopped green pepper
Note, the vegetables above can be added raw, or sauteed beforehand. Both are great, but the flavor and character changes.
Folded-in: Lay these on top of the omelet prior to folding:
Cheeses…Swiss cheese, Feta, Parmesan, Mozzarella, Cream cheese, etc. Heck even Kraft singles are great. Sliced or grated, doesn’t matter. Good cheddar tends to curdle and give up a lot of free oil, so it is the only cheese I would avoid.
Thinly sliced meat of any kind.