Heat grapeseed oil in a medium skillet over medium-high heat. Add mushrooms and cook, stirring, until softened, about 5 minutes. Remove from heat.
Place a pizza stone (available at most kitchen-supply stores) on floor of a gas oven (remove racks) or bottom rack of electric oven. Preheat oven to at least 500 degrees for 1 hour.
Shape dough into a round. Holding top edge of dough round in both hands, let bottom edge touch work surface. Carefully move hands around edge to form a circle, as if turning a wheel. Hold dough on back of your hand, letting its weight stretch it into a 12-inch round. Transfer dough to a pizza peel (or an inverted baking sheet) lightly dusted with semolina flour. Press out edges using your fingers. Jerk peel; if dough sticks, lift, and dust more flour underneath.
Drizzle olive oil over dough, leaving a 1-inch border. Arrange cheeses evenly over olive oil. Top with cooked mushrooms and thyme leaves; season with salt and pepper.
Heat oven to broil. Align edge of peel with edge of stone. Tilt peel, jerking it gently to move pizza. When edge of pizza touches stone, quickly pull back peel to transfer pizza to stone. (Do not move pizza). Broil until crust begins to bubble, 3 to 4 minutes. Reduce temperature to 500 degrees, and bake until crust is crisp and golden brown, 6 to 8 minutes more. (If not using broiler, bake pizza for 10 to 15 minutes total.) Remove pizza from oven using peel; drizzle with additional olive oil, if desired. Slice and serve.