Preheat oven to 400 degrees. Grease a 10-inch round baking dish.
Melt butter in a large skillet over medium heat; cook and stir onion and green pepper until softened, about 5 minutes. Stir in ham and continue cooking until heated through, 5 minutes more.
Beat eggs and milk in a large bowl. Stir in cheddar cheese and ham mixture; season with salt and black pepper. Pour mixture into prepared baking dish.
Bake in preheated oven until eggs are browned and puffy, about 25 minutes. Serve warm.
Servings 6