Preheat broiler.

Whisk together whole eggs, milk, and 1/2 teaspoon each of salt and pepper.

Saute onions in oil then°add vegetables in an ovenproof 12-inch nonstick skillet over medium-high heat, stirring occasionally, until softened and golden, about 8 minutes. Add meat. Add egg mixture and half of cheese and cook, lifting up cooked egg around edges to let uncooked egg flow underneath, 3 to 5 minutes (top will still be moist).

Sprinkle top with remaining cheese and broil 4 to 6 inches from heat until set, puffed, and golden brown, about 3 minutes.

Cool 5 minutes, then loosen edges with a spatula and slide onto a plate. Serve warm or at room temperature.