Cook the pasta according to package directions until al dente. Reserve about 1 cup of the starchy pasta water before draining. Set the pasta aside.

Heat olive oil in a large skillet or Dutch oven over medium-high heat. Add the sliced mushrooms and cook without stirring for 3-4 minutes to allow them to brown. Then, stir and continue cooking until they have released their liquid and are nicely caramelized (about 8-10 minutes total).

Add the minced garlic and dried thyme to the skillet. Cook for 1 minute until fragrant. If using wine, pour it in, scraping up any browned bits from the bottom of the pan, and let it simmer until it has reduced by half.

Reduce the heat to medium-low. Stir in the heavy cream and broth. Bring to a gentle simmer and let the sauce thicken slightly, about 3 minutes. Season generously with salt and pepper.

Remove the skillet from the heat. Stir in the butter until melted, followed by the grated Parmesan cheese until the sauce is smooth and creamy. If the sauce is too thick, add a splash of the reserved pasta water until the desired consistency is reached.

Add the drained pasta directly to the skillet with the mushroom sauce. Toss gently until every strand is coated. Taste and adjust seasoning one last time.

Serve immediately in bowls, garnished with fresh chopped parsley and an extra sprinkle of Parmesan cheese, if desired.