Boil salted water. Cook pasta until al dente.
Reserve 1 cup pasta water. Drain pasta. Set aside.
While pasta cooks, melt butter in large skillet over medium-high heat.
Add mushrooms. Cook 6 to 8 minutes until golden. Do not stir too much.
Add garlic. Cook 30 seconds.
Add wine or broth. Scrape up brown bits. Simmer 2 minutes.
Reduce to medium heat. Add cream, Parmesan, thyme, salt, pepper, and red pepper flakes.
Stir together. Simmer 3 to 4 minutes until sauce thickens.
Add pasta to sauce. Toss to coat.
Add reserved pasta water if sauce is too thick.
Remove from heat. Stir in parsley.
Serve immediately. Top with Parmesan, parsley, and pepper.