Preheat oven to 375 degrees. Bring a large pot of generously-salted water to a boil on the stove.

Lay the garlic cloves out on a sheet of aluminum foil, and drizzle with 1 tablespoon olive oil and sprinkle with a pinch of salt and pepper. Wrap up the edges of the aluminum foil to make an enclosed pouch, and set on the corner of a baking sheet. Spread out the mushrooms on the rest of the baking sheet, and drizzle with an additional 1 tablespoon of olive oil, as well as another pinch of salt and pepper. Roast for 25-30 minutes, until the mushrooms are browned and the garlic cloves are slightly golden. Remove from oven, open garlic pouch to cool, and set aside.

Meanwhile, heat the remaining 1 tablespoon of olive oil over medium-high heat. Once the oil is shimmering, add the shallots and cook for 4-5 minutes or until they are translucent and begin to slightly brown. Add the wine and stir for a minute, stirring the bottom of the skillet to pick up any browned bits. Add the chicken broth and thyme sprigs and cook for 5 minutes to reduce the sauce. Add the roasted mushrooms, garlic, and cream and stir to combine. Cook for an additional few minutes as sauce thickens. Once the sauce is ready, remove the thyme sprigs.

While preparing the sauce, cook the pasta al dente in the boiling water according to package instructions. Once it is ready and drained, add the pasta to the prepared sauce and toss to combine. Serve immediately and garnish with Parmesan.