Step 1Heat the oven to 450 degrees. Rub the clove of garlic all over the inside of a heavy gratin dish, then butter it well and set it aside.
Step 2Slice the potatoes and turnips as thinly as you can, ideally using a mandoline or Japanese slicing tool. Toss them with salt and place them in rough layers in the gratin dish. Don't worry about arranging them, you'll be stirring them later. Bake until the potatoes and turnips have softened, 20 to 30 minutes. Stir with a spatula every 10 minutes, making sure the bottom layer doesn't scorch.
Step 3Pour the whipping cream over the potatoes. It should come just to the top layer without completely covering it. Distribute the cheese over the top and return the dish to the oven until the cream has thickened and the top has browned, about 30 minutes. Serve hot.