Step 1 In a medium saute pan or skillet over medium heat, saute each variety of mushroom in just enough butter to keep the mushrooms from sticking. Season each batch with about one-eighth teaspoon salt and a pinch of white pepper and cook, stirring, just until the mushrooms are tender but not colored, a few minutes for each batch. When one type of mushroom is cooked, drain, turn it into a large bowl and repeat with the next type. You need to saute the mushrooms separately because each type has a different cooking time.
Step 2 Mix the cooked mushrooms together in the bowl. Stir in the garlic and thyme and set aside at room temperature until needed. (The mushrooms can be sauteed up to 2 hours ahead and kept covered with plastic wrap at room temperature.)
Step 3Place a rack in the center of the oven and heat the oven to 350 degrees. Butter a 10-inch baking dish.
Step 4Pour the cream into a large bowl and whisk in 2 1/2 teaspoons salt, one-half teaspoon white pepper and the nutmeg. (Add more salt than you might normally add because the potatoes will need it.)
Step 5Peel the potatoes and slice them into one-eighth-inch thick rounds using a mandoline, the thinnest slicing blade on the food processor or a knife. Toss the potatoes into the cream as you slice them.
Step 6Using your hands, remove enough potato slices from the cream to make a single layer on the bottom of the pan, arranging the slices in even, overlapping concentric circles. Make a second layer of potato slices and then pour some cream over the layers. Press down on the potatoes to compact the layers -- when you do this, some of the cream should rise up between the slices.
Step 7Spread the mushrooms (minus whatever liquid may have accumulated in the bowl) over the potatoes and pour in more cream, again using your hands to press down on the ingredients and bring the cream to the top. Arrange the remaining potatoes in layers over the mushrooms, pouring in cream and pressing down as you finish each layer. You may not need all of the cream; you've added enough when, without pressing down, you see cream at the edges.
Step 8Dust the gratin evenly with the Parmesan cheese and place the gratin on a foil-lined baking sheet.
Step 9Bake the gratin 45 minutes, then check that it's not getting too brown. If necessary, lower the oven temperature to 300 degrees to keep the gratin from coloring too much. Bake an additional 30 minutes or until you can easily pass a slender knife through the layers.
Step 1 0Remove the gratin from the oven and keep it warm about 20 minutes to allow the potatoes to soak up more cream. To serve, cut into wedges.