Brown butter in a large skillet over medium heat. Once browned, add mushrooms. Cook until the liquid evaporates, roughly 8 minutes.
Add minced shallot, minced garlic, and thyme leaves. Stir for 2 minutes then add beef broth. Let broth reduce by half. Roughly 12 minutes.
Add heavy cream. Stir to combine. Let reduce by half, roughly 12 minutes. Taste and add salt and pepper as needed.
Cook pasta according to package's instructions. Reserve 1 cup pasta water
Remove the mushrooms from the sauce and set aside. Strain the pasta then add it to the sauce. Stir so that each pasta is coated evenly. Add pasta water if necessary.
Transfer pasta to a serving dish. Add burrata on top and break it so that the creamy inside is open. Spoon the mushrooms on top. Drizzle with truffle oil if using. Add extra thyme to garnish. Serve and enjoy!