Preheat oven to 425 degrees.
Peel & slice potatoes 1/8” think (while you can prep potatoes by hand, a mandolin provides the easiest and fastest way to get a uniform potato slice). Here is the mandolin I have, and love. Arrange about 1/3 of the sliced potatoes in the baking dish and coat with melted butter using a pastry brush (not required but definitely the most efficient way to spread the butter in an even coat). Sprinkle salt, pepper, and shredded cheese over the first layer of potato slices, and then repeat the layers.
Bring the milk to a boil over low heat in a small saucepan; pour over the potatoes.
Bake at 425 degrees for 30-40 minutes. Let cool for about 10 minutes before serving.