Crack the eggs into a bowl. Add a few spoonfuls of milk, a generous pinch of salt, and some freshly ground pepper. Combine the ingredients with a fork until incorporated and smooth.
Over medium heat, melt a generous pat of butter in the frying pan and swirl it around to evenly coat the sides and bottom. Pour in the egg mixture.
As the eggs start to set, use a rubber spatula to gently push aside a small patch of cooked egg, then tilt the pan so that uncooked egg fills in the gap. Repeat, working your way around the pan until the eggs have reached the consistency you like.
Sprinkle grated cheese evenly over the cooked eggs. At this point you can also add any other fillings you might want, such as fresh herbs, avocado, smoked salmon, cooked ham, sun-dried tomatoes, or roasted bell peppers. Turn off the burner.
Using the rubber spatula, lift up one edge of the omelet and roll it over itself about one-third of the way. Hold the pan over a plate and let the omelet start to slide onto it, flipping the pan at the last minute so the omelet folds over itself again.