Steam 4 cups broccoli florets until crisp-tender. Rinse under cold water. Drain.

Heat oil in heavy large skillet over medium-high heat. Add mushrooms and garlic and saute until mushrooms are almost tender, about 5 minutes. Add cream, thyme and red pepper and bring to boil. Add pasta and broccoli and cook until pasta is coated, tossing gently, about 3 minutes. Season with salt and pepper. Transfer to large bowl. Sprinkle with Parmesan.