Break the eggs into a bowl. Add the milk, if using. Season with salt and pepper and whisk until evenly blended.
Have a serving plate ready by the stove.
Heat the butter in a small nonstick omelet pan over high heat, tilting the pan to coat the entire surface.
Pour the egg mixture into the pan and scramble it with the back of a fork or wooden spoon as you gently shake the pan. Keep moving the pan and utensil at the same time until the egg mixture has coagulated and is nearly set, 2–3 minutes. Add the cheese and gently press it down into an even layer.
Lower the heat and let the egg mixture finish cooking without stirring, 15–20 seconds.
Tilt the pan and slide a fork or spoon around the lip of the pan, under the omelet, to be sure it is not sticking. Slide the omelet to the front of the pan and use a fork or a wooden spoon to roll the edge nearest the handle toward the center of the omelet.
Tip the pan, rolling the omelet onto the serving plate. To give the omelet additional sheen, rub the surface lightly with butter.
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