Bring a large pot of water to a boil and season generously with salt. Once boiling, add the pasta and cook, stirring occasionally, until al dente. Reserve ½ cup of the pasta water and drain.

In a large skillet, melt the butter over medium heat. Cook, swirling occasionally, until the butter starts to foam and smells slightly nutty, about 3 to 4 minutes. Add the mushrooms and cook, stirring occasionally, until tender and browned, about 5 minutes. Add the garlic and thyme, and cook for 2 more minutes. Season with salt and pepper to taste.

Add the pasta to the skillet and stir in the reserved pasta water. Reduce heat to low and add half of the Parmesan cheese, stirring gently until melted. Add the remaining cheese and stir until melted. Use tongs to toss the pasta and cheese to prevent clumping. Taste and season with salt and pepper if necessary.

Transfer the pasta to plates or bowls and garnish with parsley, crushed red pepper flakes, and extra Parmesan cheese, if desired.