Bring a large pot of water to a boil. Add 2 tablespoons kosher salt, then the pasta. Cook 2 minutes less than package directions for al dente. Reserve 1 cup pasta water, then drain.

Heat 2 tablespoons olive oil in a large skillet over medium-high heat until shimmering. Add half the mushrooms in a single layer. Cook undisturbed until deeply browned on the bottom, about 4 minutes. Flip and cook 3 minutes more. Transfer to a plate. Repeat with remaining mushrooms.

Reduce heat to medium. Add butter and remaining 1 tablespoon olive oil to the skillet. Add garlic and cook until golden at the edges, about 2 minutes. Stir in thyme and red pepper flakes, then pour in wine. Simmer until reduced by half, about 3 minutes.

Return mushrooms and any juices to the skillet. Add drained pasta, 1/2 cup reserved pasta water, and Parmesan. Toss vigorously over medium heat until a creamy sauce coats the pasta, about 2 minutes. Add more pasta water if needed. Remove from heat and stir in parsley. Season with black pepper.

Divide among warm bowls. Top with additional Parmesan and a drizzle of olive oil if desired.