- Preheat an oven to 375°F. Butter an 8-inch square baking dish.
- In a saucepan, combine the cream, garlic, thyme sprigs, and a pinch each of salt and pepper. Bring to a low boil over medium heat, then remove from the heat and set aside.
- Peel the potatoes and, using a mandoline or a very sharp knife, cut them into slices about 1/8 inch thick. Gently stir the potato slices into the cream mixture, cover and let stand while you cook the mushrooms.
- In a fry pan over medium heat, warm the olive oil and melt the 1 tablespoon butter. Add the mushrooms and a pinch of salt, and sauté until all of the liquid released by the mushrooms has evaporated, 7 to 9 minutes. Add the minced thyme with a pinch of pepper and cook for 1 minute.
- Arrange one-third of the potato slices, slightly overlapping, on the bottom of the prepared dish. Sprinkle lightly with salt and pepper and 1 tablespoon of the cheese. Spread half of the mushrooms over the potatoes and sprinkle with 1 tablespoon of the cheese. Repeat these layers, using half of the remaining potatoes and all of the remaining mushrooms, and sprinkling with salt, pepper and 1 tablespoon cheese between the layers of vegetables. Top with the remaining potatoes and sprinkle with salt and pepper. Using a large, flat spatula, gently press on the vegetables to compact them. Pour the cream mixture through a strainer into the dish and sprinkle with the remaining cheese.
- Cover the dish with aluminum foil and bake until the potatoes are tender when pierced with a sharp knife, about 45 minutes. Remove the foil and bake until the mixture is golden brown and bubbly, about 20 minutes more. Let the gratin rest for about 10 minutes, then use a sharp-bladed spatula to cut it into squares and serve immediately.