Beat eggs vigorously with a fork until a mousse of tiny bubbles froths along the surface. While you heat a small nonstick pan over medium-high heat, sprinkle salt and pepper on eggs.
When pan is very hot, add oil, then butter; tilt pan to coat surface evenly. Pour eggs into center. As they start to firm up, run a spatula (or the underside of a fork) under the edges, bunching mixture toward the center so the remaining liquid can cook.
When egg is firm and nearly dry, fold over omelet with spatula. (With practice, you can fold the omelet by flicking the pan forward, then back.) Slide omelet onto a plate. Serve immediately.