Preparation

• Heat the oven to 375°F and butter a 9×13-inch baking dish with the softened butter.
• Prepare the mushroom sauté, using the optional deglazing liquid if you like. Transfer the mushrooms to a bowl and return the pan to medium-high heat. Pour in the broth and scrape the pan with a wooden spoon until most of the cooked-on bits are released into the broth. Combine the broth with the cream in a measuring cup with a spout.
• Peel and slice the potatoes very thinly (about 1/8-inch thick). In a large bowl, toss the sliced potatoes with the thyme, salt, and several grinds of pepper. Overlap half of the potato slices in two even layers to cover the bottom of the buttered baking dish. Distribute the mushrooms over the potatoes. Cover the mushrooms with two more overlapping layers of the remaining potato slices. Pour the broth and cream mixture over the potatoes. Sprinkle the cheese evenly over the top.
• Cover the gratin with foil and bake it for 30 minutes. Uncover it and press the potatoes down with a spatula. Bake uncovered, rotating the pan occasionally for even browning, until the top is browned and the potatoes feel tender when poked with a fork, another 25 to 30 minutes. The cream will be gently bubbling. Let the gratin sit for about 15 minutes before serving.