Diner-Style Western Omelet

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.