Brush 13x9x2-inchglass or oval ceramic baking dish with2 tablespoons oil. Arrange 1/3 of potatoes,slightly overlapping, in dish.
Sprinkle with1/2 teaspoon coarse salt and 1/4 teaspoonpepper.
Pour 1/3 cup cream over; sprinklewith 1/4 cup cheese. Repeat layering 1/3 ofpotatoes, 1/2 teaspoon salt, 1/4 teaspoonpepper, 1/3 cup cream, and 1/4 cup cheese 2more times.
Bake uncovered until potatoesare tender, adding cream by tablespoonfulsif dry, about 45 minutes.
Remove from oven;maintain oven temperature.
Sprinkle thyme and garlic slices overgratin. Toss mushrooms in medium bowlwith 3 tablespoons oil; sprinkle with coarsesalt and pepper. Arrange mushroom slicesatop gratin around edge of dish.
Drizzlewith 1/4 cup cream; sprinkle with 1/4 cupcheese. Continue to bake uncovered untilmushrooms are tender and potato edgesare golden brown, about 20 minutes longer.DO AHEAD: Can be made 1 day ahead.Cool 1 hour in dish; cover and chill. Beforeserving, cut gratin into 10 pieces.
Transferto rimmed baking sheet. Cover with foil andrewarm in 350°F oven, 10 to 15 minutes.