Crank the oven to 350°F (175°C).
Rub the crushed garlic clove all over the inside of a 2 1/2-quart casserole or a 9-by-13-inch baking dish, then butter it thoroughly.
Pour the cream (or cream and milk) into a saucepan. Add the garlic clove, thyme sprigs, a few grinds of nutmeg, and a large pinch of salt.
Warm the cream over low heat and gently simmer for 10 minutes. Remove the pan from the heat and fish out the garlic and thyme.
Lay a single layer of sliced potatoes in the baking dish, overlapping the slices like playing cards. Sprinkle with a small pinch of salt and a grind of fresh pepper.
Continue layering and seasoning the potatoes until all of the potatoes are used.
Pour the seasoned cream over the potatoes and sprinkle the top with the cheese.
Cover the dish with aluminum foil and bake for 1 1/2 hours.
Remove the foil and continue baking until the top is bubbling and spotted with brown, and a fork easily pierces the potatoes, 25 to 35 minutes, depending on the size and depth of your dish. Let stand for 15 minutes before serving.