Preheat the oven to 350 degrees, F. Rub a 9 X 12-inch gratin dish with the garlic clove, then with 2 T of butter.
Heat the butter in a medium skillet, and sauté each variety of mushroom separately in just enough butter to keep the mushrooms from sticking. Season each batch with salt and a pinch of white pepper, and cook - stirring - until the mushrooms are tender. Remove the cooked mushrooms to a bowl, add the chopped garlic and thyme, and stir to combine. Set aside.
Pour the cream into a large bowl, and whisk in salt, white pepper and the nutmeg. (Add more salt than you might normally; 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 the potatoes in the 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 and 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, and place the gratin dish on a foil-lined baking sheet.
Bake the gratin uncovered until all liquid is absorbed and the potatoes are tender and golden, about 1 1/2 hours. Half-way through cooking, check that it's not browning too much. If so, lightly cover the gratin with foil. Remove the gratin from the oven, and keep it warm for about 20 minutes to allow the potatoes to soak up the cream and thicken.