Preheat the oven to 400 degrees F (200 degrees C). Grease a 10-inch round baking dish.

Melt butter in a large skillet over medium heat. Add onion and bell pepper; cook and stir until softened, about 5 minutes. Stir in ham; continue cooking until heated through, about 5 minutes.

Beat eggs and milk together in a large bowl; stir in Cheddar cheese and ham mixture. Season with salt and black pepper. Pour into the prepared baking dish.

Bake in the preheated oven until eggs are browned and puffy, about 25 minutes. Serve warm.