Press hash brown potatoes between paper towels to remove excess moisture. Transfer potatoes to an 8-inch pie dish and press into the bottom and up the sides of the dish to form a crust. Drizzle with melted butter.
Bake in preheated oven until golden, about 25 minutes. Remove from oven and reduce heat to 350 degrees F (175 degrees C).
Combine ham, Cheddar cheese, onion, and bell pepper in a bowl; spread mixture over potato crust. Beat eggs, milk, salt, and black pepper in the same bowl; pour over ham mixture.
Bake in oven until eggs are set, 25 to 30 minutes.
