Bring a large pot of salted water to a boil.
Cook pasta until al dente. Reserve 1/2 cup cooking water, then drain.
Heat olive oil and butter in a wide skillet over medium-high.
Add mushrooms and sear, stirring occasionally, until deeply browned, 6–8 minutes.
Stir in shallot and cook until softened, about 2 minutes.
Add garlic and thyme; cook until fragrant, 30 seconds.
Deglaze with white wine, scraping up browned bits. Simmer to reduce by half.
Pour in broth and cream. Simmer until slightly thickened, 3–4 minutes.
Lower heat and slowly melt in Gruyere. Add lemon juice, red pepper flakes, salt, and pepper.
Toss in pasta. Splash in pasta water as needed to coat and gloss.
Finish with parsley and Parmesan. Serve hot.