Preheat oven to 425°. In a medium bowl, toss together Swiss, Parmesan, parsley, thyme, and garlic powder.
Arrange one-third of mushrooms on the bottom of a 2-quart baking dish. Season with salt and pepper. Sprinkle with one-third of the cheese mixture. Repeat layering two more times. Pour heavy cream over mushrooms.
Cover with foil and bake until mushrooms are tender about 35 minutes.
Remove foil and carefully pour off any excess liquid in the bottom of the dish. Broil until top is golden, about 4 minutes more.