Melt the butter in an 8-inch nonstick skillet over medium heat.
In a small bowl, lightly beat the eggs, salt, and a couple of grinds of pepper with a fork until the whites and yolks are completely mixed and the eggs are frothy. Pour the eggs into the skillet and let them sit, undisturbed, until the eggs are just beginning to set around the edges, 30 to 60 seconds.
Gently scramble the eggs with a silicone spatula while shaking the pan back and forth. Scramble and shake just until the eggs have set on the bottom but are still undercooked on top, 45 to 60 seconds. Turn off the heat and give the pan a few shakes to evenly distribute the eggs. Tap it once firmly on the stove to smooth the bottom of the omelet (use a heatproof cutting board if your stovetop is glass).
Sprinkle the cheese down the center third of the omelet and run a spatula around the edge. You can use the spatula to smooth the top of the eggs if necessary. Let it sit for about 1 minute. The top of the omelet will still be loose and the bottom should have no color.
Fold a third of the omelet over the cheese. Shake the unfolded third of the omelet up the side and slightly out of the pan. Invert the pan onto a plate to complete the final fold of the omelet. It should be folded in three, like a letter. Rub the top with cold butter for shine, if desired. Serve immediately.