Gather all ingredients.

Beat eggs in a small bowl until just combined; do not overbeat.

Melt butter in a skillet over medium-high heat. Add ham, onion, and bell pepper; season with salt and pepper. Cook and stir until onions soften and ham begins to caramelize, about 5 minutes.

Reduce heat to medium-low and pour in eggs. Mix briefly with a spatula while shaking the pan to ensure ingredients are evenly distributed.

Quickly run the spatula along the edges of the omelet.

Sprinkle Cheddar cheese and cayenne pepper over the omelet.

Cook, shaking the pan occasionally, until top is still wet but not runny, about 5 minutes. Use a spatula to fold omelet in half and transfer it to a plate.