Bring a large pot of well-salted water to a boil. Add the pappardelle and cook until al dente according to the package directions. Reserve about 1 cup of pasta water, then drain the pasta.
While the pasta cooks, heat olive oil in a large skillet over medium-high heat. Add the mushrooms in an even layer and cook undisturbed for 3–4 minutes to brown. Stir and continue cooking until the mushrooms are deeply golden and their liquid has mostly evaporated, about 5–7 minutes more. (Cook in batches if the pan is crowded.)
Add butter to the skillet. Once melted, stir in the minced garlic and cook for 30–60 seconds until fragrant.
If using, pour in the white wine and simmer for 1–2 minutes, scraping up browned bits, until mostly reduced.
Lower heat to medium. Stir in the heavy cream and simmer for 2–3 minutes to slightly thicken.
Reduce heat to medium-low and stir in the Parmesan and mozzarella until melted and smooth. Season with black pepper and salt to taste.
Add the drained pappardelle to the skillet and toss to coat. Splash in reserved pasta water a little at a time until the sauce clings to the noodles and looks glossy.
Finish with chopped parsley and extra Parmesan. Serve immediately while hot and creamy.