Heat the oven to 425 degrees. Put the milk in a small heavy saucepan and peel and smash one of the garlic cloves. Add it to the pot and then heat the milk gently until it starts to bubble at the edges. Remove from the heat and let steep while you continue with the recipe.
Peel the second garlic clove, cut it in half and rub the cut side around the inside of an oval gratin dish about 9 inches long and 2 inches deep. Rub 1 tablespoon of the butter over the inside of the baking dish.
Peel the potatoes and cut them into 1/8-inch-thick slices (I use a mandoline to get them nice and even), laying the slices on a kitchen towel to drain. Layer about a third of the potatoes in the bottom of the baking dish, fanning them into concentric, overlapping circles. Season the potatoes with salt and pepper and sprinkle a third of the cheese over them. Repeat with two more layers of potatoes, salt and pepper and cheese, making the top layer as neat and tidy as you can.
Remove the garlic clove from the hot milk and pour the milk evenly over the potatoes. Dot the top of the potatoes with the remaining tablespoon of butter and bake the gratin for about 30 minutes, until it's browned and bubbly. Let the potatoes cool for 5 minutes before serving.