Cook the Pasta: Bring a large pot of well-salted water to a boil. Add pasta and cook until just shy of al dente (about 8–10 minutes). Reserve ½ cup pasta water, then drain.
Sear the Mushrooms: In a cold skillet over medium-high heat, melt 1 Tbsp butter with 1 Tbsp olive oil. Add oyster mushrooms in a single layer; cook undisturbed for 2–3 minutes until golden, then stir and cook another 2 minutes. Transfer to a plate.
Sauté Aromatics: Reduce heat to medium. Add remaining 2 Tbsp butter. Sauté diced onion until translucent, about 3–4 minutes. Stir in garlic and cook 30 seconds. If using, sprinkle flour and cook 1 minute.
Build the Sauce: Whisk in broth, scraping brown bits. Pour in cream and bring to a gentle simmer. Season with thyme, salt, pepper, and red pepper flakes. Simmer until slightly thickened, about 4–5 minutes.
Combine & Finish: Return mushrooms to the pan. Add drained pasta and toss to coat. If needed, loosen sauce with reserved pasta water. Off the heat, stir in cheese and lemon zest.
Garnish & Serve: Plate the pasta, sprinkle parsley on top, and finish with a drizzle of extra-virgin olive oil and extra black pepper.