Beat eggs until just combined; do not over-beat.
Melt butter in a skillet over high heat. Add ham, onion, and bell pepper. Sprinkle in a nice pinch of salt and a touch of pepper. Cook and stir over medium-high heat until onions start to soften and ham begins to caramelize, about 5 minutes.
Reduce heat to medium-low and pour in the 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 the top is still wet but not runny, about 5 minutes. Fold omelet in half with your spatula and transfer to a plate.
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