Preheat oven to 350 degrees F (175 degrees C). Grease a 9×13-inch baking dish.
Arrange hash brown patties on the prepared baking dish. Sprinkle Cheddar cheese and ham over patties.
Beat eggs, milk, salt, and mustard in a large bowl using a whisk; pour over ham. Cover baking dish with aluminum foil.
Bake in the preheated oven for 1 hour. Remove foil and bake until edges are golden brown and a knife inserted in the center comes out clean, about 15 minutes more.
Substitute 2 cups bacon or sausage for the ham if desired.
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