Adjust oven rack to middle position and heat oven to 400 degrees. Combine cheeses in a large bowl. Transfer 1/3 of cheese mixture to a separate bowl and set aside. Add cream, garlic and thyme to cheese mixture. Season with salt and pepper. Toss potato slices until every slice is coated with cream mixture.

Grease a 2-quart casserole dish with butter. Organize potatoes into a stack, laying them in the dish with their edges aligned vertically. Continue placing potatoes until all are added. Pour excess cream/cheese mixture over potatoes until the mixture comes halfway up sides of the casserole.

Cover dish tightly with foil and transfer to the oven. Bake for 30 minutes. Remove foil and continue baking until pale golden brown, about 30 minutes longer. Sprinkle with remaining cheese and return to oven. Bake until deep golden brown and crisp on top, about 30 minutes longer. Let rest for a few minutes, and serve.