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 until golden. 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. Simmer 2 to 3 minutes until slightly thickened.
Toss drained pasta into the sauce. Add pasta water as needed for a silky consistency. Stir in parmesan until melted.
Return mushrooms to the pan. Add spinach and fold until just wilted, about 1 minute. Serve with extra parmesan.