Preheat oven to 375 degrees.
Peel the potatoes and cut into slices somewhere between 1/8 and 1/16 inch thick. The slicing disk of a food processor does this rapidly. Drop the slices in cold water as you go.
Select a shallow baking dish (an oval gratin 8x14x2 inches is ideal) and grease it well with 1 tablespoon of the butter. Crush the garlic cloves and use one to rub the dish all around.
Put the milk in a saucepan, add all the garlic cloves, including the one used for the dish, and several gratings of nutmeg. Bring to a simmer. Meanwhile, drain the potatoes well and layer them in the prepared dish, seasoning each layer with salt and pepper. Strain the milk over, sprinkle with the grated cheese, and dot with butter. Bake for about 1 hour, or until the milk is absorbed and the top is nicely browned. Remove from the oven and let set for 10 or 15 minutes before serving.
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