Heat oil in a large nonstick skillet over medium heat. Add onion; sauté 5 minutes. Add mushrooms, thyme, salt, and pepper; sauté 5 minutes. Stir in broth, vermouth, and garlic; reduce heat to medium-low, and cook 5 minutes.
Combine the mushroom mixture and the pasta in a large bowl, and toss well. Sprinkle each serving with cheese and chopped chives.