Combine the eggs, milk, salt, and pepper in a medium bowl and mix very well with a fork or a whisk.
Set a plate by the stove. Heat a small (about 6 inches) seasoned omelet pan or non-stick skillet over medium-high heat. When the pan is warm, add the butter (it should sizzle gently). Swirl the pan to distribute the butter as it melts. When the butter stops sizzling and the foam subsides add the eggs. Pause to let the eggs heat slightly and then stir vigorously, with a heatproof spatula, making sure you include the sides of the egg mixture occasionally so the omelet cooks evenly. Once the eggs are just set, bang the pan gently on the burner to release the omelet from the pan. Check to see that it is not sticking to the sides or bottom of the pan, if so release the omelet with a heat proof spatula. Hold the pan at a 45-degree angle to the stove and carefully fold the omelet like a business letter. Cook just until the desired degree of doneness lifting the pan or reducing the heat to prevent browning. (A classic omelet doesn't have any browning on it.) Transfer to a warm plate and serve.
Omelet Filling Suggestions: There are many omelet fillings, both raw and cooked, sweet and savory. Here is a starting point, but feel free to improvise. Once the eggs are set add any of these ingredients before folding. An alternative method, to use with delicate fillings, like creme fraiche and herbs, fresh fruits or caviar; is it to roll the omelet and then slice the top open, as you would a baked potato, and then fill it.
- 1 tablespoon finely chopped fresh herbs, like Italian parsley, basil, dill, tarragon, chives, thyme, and chervil. Use the herbs alone or in combination, like the classic combination fine herbs (equal parts parsley, chervil, chives and tarragon.) Brush the finished omelet with butter and sprinkle with additional herbs. Herbs can also be added to the omelet mixture to flavor the eggs more fully.
- 2 to 3 tablespoons cooked vegetables like, chopped asparagus, spinach or other greens, zucchini, mushrooms, eggplant or peppers