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Preparation Heat oil in a medium (preferably 10") nonstick skillet over medium. Cook ham, onion, and bell pepper, stirring often, until softened and beginning to brown, 5–7 minutes.
Meanwhile, whisk eggs, milk, salt, and pepper in a medium bowl.
Reduce heat to medium-low and shake pan so that ham and vegetables form a single layer. Pour egg mixture over ham and vegetables, then sprinkle cheese over. Cook, tilting skillet and gently running a rubber spatula around edges to allow uncooked egg to flow underneath, until eggs are mostly set, about 5 minutes. Fold omelet in half, cover, and cook until set, about 1 minute. Slide onto a plate to serve.