Working in a medium sauté pan or skillet over medium heat, sauté each variety of mushroom in just enough butter to keep the mushrooms from sticking. Season the mushrooms with salt and pepper and cook, stirring, just until they are tender but not colored, a few minutes for each batch.
When one type of mushroom is cooked, drain, turn it into a bowl, and repeat with the next type. You need to sauté the mushrooms separately because each type has a different cooking time, but, once cooked, all the mushrooms should be mixed together in the bowl, along with the garlic and thyme. Set aside at room temperature until needed. (The mushrooms can be sautéed up to 2 hours ahead and kept covered with plastic wrap at room temperature).
Center a rack in the oven and preheat the oven to 350 degrees F. Butter the bottom and sides of an ovenproof 10-inch sauté pan or skillet.
Pour the cream into a large bowl and whisk in salt and pepper and freshly grated nutmeg to taste. (Add more salt than you might normally, because the potatoes will need it.) One at a time, peel the potatoes and slice them into 1/8-inch-thick rounds. The best way to get rounds this thin is to use a mandolin. Lacking that, use the thinnest slicing blade on the food processor or a sharp knife. Toss the potatoes into the cream as you slice them.
Using your hands, pull enough potato slices out of the cream to make a single layer on the bottom of the buttered pan, arranging them in even, slightly 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. Spread the mushrooms (minus what ever liquid they 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 attractive layers over the mushrooms, pouring in cream and pressing down as you finish each layer. You may not need all of the cream – you’ll know you’ve added enough when, without pressing down, you see cream at the edges of the pan.
Dust the gratin evenly with the Parmesan cheese and place the pan on a foil-lined baking sheet that can act as a drip-catcher.
Bake the gratin for 45 minutes, then check that it’s not getting too brown. If necessary, lower the oven temperature to 300 degrees F to keep the gratin from coloring too much. Bake for 15 minutes more, or until you can easily pass a slender sharp knife through all the layers.
Remove the gratin from the oven and let it stand in a warm place for about 20 minutes, time enough for the potatoes to soak up more cream. If it’s more convenient, you can keep the gratin warm in a 200 degree F oven. Bring the gratin to the table and cut it into wedges.