Heat butter and olive oil in a large, deep skillet or pot over medium-high heat. Add sliced mushrooms and cook for 6-8 minutes until golden brown and their moisture has evaporated. Don't stir too often—let them get nice and caramelized!
Add chopped shallot and cook for 2 minutes until softened. Stir in garlic and thyme, cooking for another minute until fragrant.
If using white wine, pour it in now and let it simmer for 2 minutes to cook off the alcohol.
Add the pasta and broth to the pot, making sure the pasta is mostly submerged. Season with salt and pepper. Bring to a boil, then reduce to a simmer.
Cook for 10-12 minutes, stirring frequently, until pasta is al dente and most of the liquid has been absorbed.
Reduce heat to low and stir in heavy cream, Parmesan cheese, and nutmeg (if using). Stir until the sauce is creamy and the cheese has melted completely.
Taste and adjust seasoning if needed. Garnish with fresh parsley and extra Parmesan. Serve immediately and watch it disappear!