Heat oil over medium heat in a 10-inch skillet. Add broccoli, sweet pepper, onion, Italian seasoning, salt, and pepper. Cook and stir until florets are crisp-tender, about 4 minutes for fresh broccoli or about 5 minutes for frozen broccoli.
Combine eggs and milk in a medium mixing bowl; beat well. Pour over vegetable mixture. As the eggs begin to set, run a spatula around the edge of the skillet, lifting the egg mixture to allow the uncooked portions to flow underneath. Continue cooking and lifting edges until eggs are nearly set. (The surface will be moist.)
Remove skillet from heat; sprinkle with cheese. Cover; let stand for 3 to 4 minutes or until set. To serve, cut into wedges. Makes 6 main-dish servings.