Cook pasta in boiling salted water until al dente. Drain, reserving 1/2 cup cooking water. In large skillet, cook onion, carrot, garlic, mushrooms, lamb, and thyme in oil over medium-high, stirring often, until lamb is browned, 10 minutes; season. Stir in wine, then tomatoes. Simmer over medium for 10 minutes. Toss in pasta, cooking water, and cheese.