Bring a large pot of salted water to a boil. Once the water is boiling, add the pasta and cook until al dente, stirring occasionally. Before draining, reserve ½ cup of the pasta cooking water, then drain the pasta.

In a large skillet over medium heat, melt the butter. Let it cook, swirling the pan occasionally, until it begins to foam and gives off a nutty aroma—this should take about 3 to 4 minutes. Add the sliced mushrooms and cook, stirring every so often, until they're browned and tender, about 5 minutes. Stir in the chopped garlic and dried thyme, and cook for an additional 2 minutes. Season with kosher salt and freshly ground black pepper to taste.

Add the cooked pasta to the skillet and pour in the reserved pasta water. Turn the heat to low and mix in half of the Parmesan cheese, stirring gently until it melts. Add the remaining cheese and continue stirring until smooth and melted. You can also use tongs to toss the pasta and cheese together, helping to avoid clumping. Taste and adjust seasoning with more salt and pepper, if needed.

Serve the pasta in bowls or on plates, and garnish with chopped parsley, a pinch of crushed red pepper flakes, and extra Parmesan cheese, if you like.