Cook pasta in salted water according to package directions. Reserve at least ½ cup pasta water, drain, and set aside.
Melt butter in a large high-sided skillet over medium heat. Add mushrooms and cook 4–6 minutes until browned and tender.
Add garlic and thyme; cook 1 minute more.
Sprinkle flour over mushrooms and cook, stirring, for 1 minute.
Add white wine and cook 1 minute to reduce.
Reduce heat to medium-low. Stir in broth, cream, and parmesan. Cook, stirring constantly, until thickened.
Season with nutmeg, salt, and pepper.
Add pasta and toss to combine, using pasta water to loosen the sauce as needed.
Notes
Once broth & cream are added, stir constantly. This helps thicken the sauce and prevent scorching.
Reserve a bit of pasta water. Pasta water is called 'liquid gold' for a reason, and is great for thinning out a pasta sauce without causing it to break.