Melt the butter in an omelet pan or a nonstick 7-inch pan with sloping sides. Tilt the pan from side to side to coat the bottom with melted butter. When butter is about to turn brown, add the eggs to the skillet. Over high heat, stir the eggs with your right hand while you shake the pan back and forth, up and down, with your left hand. When the underside of the omelet begins to set, tilt the pan, after you lift the edge of omelet, so that any uncooked egg from the top will run under the cooked egg and set. Run a line of the grated cheese down the middle of the omelet.