Heat the oven to 400°F and grease a 9-by-13-inch baking dish.
Toss the hash browns with the melted butter, 1/2 teaspoon salt and 1/4 teaspoon pepper and press them firmly across the bottom of the dish.
Bake the hash brown base for 20 minutes, until dry on top and browned around the edges.
Scatter the ham, bell pepper, onion and 1 1/2 cups cheddar over the hot potatoes.
Whisk the eggs, milk, remaining salt and pepper and pour the mixture evenly into the dish.
Add the remaining cheddar and bake for 28 to 33 minutes, until the center reaches 160°F and a knife comes out without wet egg.
Rest for 10 minutes before slicing.