Cook pasta in salted water until 1 minute shy of al dente. Reserve 1/2 cup pasta water before draining.
Sear mushrooms in 1 tbsp olive oil over medium-high heat without stirring for 3 to 4 minutes. Flip and cook 2 more minutes. Remove and set aside.
Cook shallot in remaining oil and butter over medium heat for 2 minutes. Add garlic and thyme, cook 30 seconds.
Add heavy cream and simmer 2 to 3 minutes until slightly thickened.
Toss drained pasta into the sauce with pasta water as needed. Remove from heat and stir in parmesan.
Return mushrooms to pan. Fold in spinach until just wilted. Season and serve with extra parmesan.