In a 12" sauté pan heat 1 TBS butter and oil over medium heat until the butter foams. Add the mushrooms and garlic. Sprinkle with 1/2 tsp salt and stir with wooden spoon until the mushrooms start to release their moisture and begin to shrink, 2-3 minutes. Increase the heat to medium high so that you hear a steady sizzle; stir occasionally. In about 5 minutes, when the liquid evaporates and the mushrooms start to brown, give just an stir with the spoon to allow the mushrooms to brown nicely, cooking them another 2-4 minutes. Turn off the heat and toss the mushrooms with parsley and a few grinds of pepper.
Transfer the mushrooms to a bowl and return the pan to med-hi heat. Pour in the broth and scrape the pan until most of the browned bits are released into the broth. Combine the broth with the cream in a measuring cup; set aside.
In a large bowl, toss the potatoes with thyme, 2 1/2 tsp salt and several grinds of pepper. Overlap half the potato slices in two even layers to cover the bottom of the prepared dish. Distribute the mushrooms over the potatoes. Cover the mushrooms with two more overlapping layers of the remaining potatoes. Pour the broth and cream mixture over the potatoes. Sprinkle evenly with cheese.
Cover the gratin with foil and bake 30 minutes. Uncover 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 are tender when tested with a fork. Let the gratin sit for about 15 minutes before serving.