Preheat oven to 325 degrees with a rack set in the lower third of oven. Combine sliced potatoes and milk in a large saucepan over high heat. Bring to a boil and immediately reduce heat to low; cover and simmer until potatoes are just tender, about 3 minutes.

Place a colander over a large bowl and drain potatoes, reserving milk. Add heavy cream to milk and stir to combine.

Rub a 2 1/2-quart oval baking dish with garlic and 1 tablespoon butter. Arrange sliced potatoes, sprinkling cheese between the layers, in the baking dish; season with salt and pepper. Dot with remaining 2 tablespoons butter and pour reserved cooking milk and cream mixture over.

Transfer baking dish to oven and bake until cheese becomes deep golden brown and milk has reduced and thickened, 80 to 90 minutes. Remove from oven and let rest for 5 to 10 minutes before serving.