Preheat the oven to 425 degrees F (220 degrees C).

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 to form a crust. Drizzle with melted butter.

Bake in the preheated oven until golden, about 25 minutes. Remove from the oven and reduce heat to 350 degrees F (175 degrees C).

Combine ham, Cheddar cheese, onion, and bell pepper in a bowl; spread over hash brown crust. Beat eggs, milk, salt, and black pepper in the same bowl; pour over ham mixture.

Bake in the oven until eggs are set, 25 to 30 minutes.