Preheat the oven to 350°F. Rub a 9-by-12-inch gratin dish with the whole garlic clove, then coat with 2 tablespoons of the butter.

In a medium skillet, sauté each variety of mushroom separately in just enough of the remaining butter to keep the mushrooms from sticking. Season each batch with salt and a pinch of white pepper and sauté just until the mushrooms are tender. Remove the cooked mushrooms to a bowl. When all the mushrooms are cooked, add the chopped garlic and thyme, and stir to combine. Set aside.

Pour the cream into a large bowl and whisk in salt and white pepper to taste and the nutmeg. (Add more salt than you might normally do; the potatoes will soak it up.) Peel the potatoes and slice them very thinly, adding the potatoes to the seasoned cream as you slice.

Layer half of the potatoes in the prepared dish, then pour a little cream over the layers. Lightly press down on the layers. Drain any excess liquid from the mushrooms and spread the mushrooms over the potatoes. Pour in a bit more cream. Cover with the remaining potatoes in layers over the mushrooms, pouring in a little cream as you go. Add enough of the cream to come up to the last layer of potatoes. Dot with butter. Place the dish on a foil-lined baking sheet.

Bake the gratin uncovered until all the liquid is absorbed and the potatoes are tender and golden, about 1 ½ hours. Halfway through cooking, check that the gratin is not getting too brown. If it is, lightly cover the gratin with foil to keep it from coloring too much.

Remove the gratin from the oven. Keep it warm for about 20 minutes to allow the potatoes to soak up the cream, then serve.