Bring 2 cups salted water to boil in medium saucepan. Add quinoa, reduce heat to medium-low, cover, and simmer until tender and water is absorbed, about 13 minutes.
Meanwhile, heat oil in large skillet over medium-high heat. Add onion and sauté until onion begins to brown, 5 minutes. Add garlic; stir 30 seconds. Add mushrooms and thyme. Sautée; until mushrooms are tender, 6 minutes. Add wine; stir until wine is reduced and liquid is syrupy, 2 minutes.
Mix quinoa into mushroom mixture; season with salt and pepper. Pass cheese separately.
