Break eggs into a small bowl; season with salt and pepper. Using a fork, beat no more than 30 strokes so only a minimal amount of foam develops. (Overbeating reduces the eggs' volume, resulting in a "skinny" omelet. The keys to success are speed and heat: Beating the eggs quickly and cooking them over high heat produces large, coagulated curds, resulting in a fluffy, soft omelet.)
Melt butter in an 8-inch nonstick skillet over medium heat until butter begins to sizzle. Raise the heat to high, and add the beaten eggs. Count to five, then raise the pan so it forms a 45-degree angle to the burner. Rapidly shake the pan back and forth so the egg mixture is thrown against the pan's sides and begins to coagulate.
Return pan to burner. Rotate pan so any uncooked egg in the center moves to perimeter of pan. Cook egg evenly, 15 to 20 seconds.
Quickly place the filling in the center of the omelet. Fold the omelet over the filling by tilting the pan a little higher off the burner and using a spatula to fold the omelet. Turn the omelet onto a plate, and serve.