Bring a large pot of salted water to a boil. Cook the tortellini according to package directions until they float or are tender. Reserve about 1 cup of the pasta cooking water before draining the tortellini. Set the drained tortellini aside.
While the water heats, heat the olive oil and butter in a large skillet over medium heat. Add the sliced mushrooms and cook until they release their moisture and begin to brown, about 6 to 8 minutes.
Add the minced garlic, thyme, salt, and pepper to the skillet. Cook for 1 minute until the garlic is fragrant.
Pour in the heavy cream and broth. Bring the mixture to a gentle simmer, stirring occasionally. Reduce the heat to low.
Gradually stir in the grated Parmesan cheese until the sauce is smooth and thickened. Do not let the sauce boil once the cheese is added.
If the sauce is too thick, add a few tablespoons of the reserved pasta water until you reach your desired consistency. Taste and adjust salt if needed.
Add the cooked tortellini to the skillet with the sauce. Toss gently to coat everything evenly. Cook for 1 to 2 minutes to heat the tortellini through.
Serve immediately, garnished with fresh parsley and extra Parmesan cheese.
Notes
If using frozen tortellini, you may need to increase the simmering time slightly after adding them to the sauce to ensure they are heated through.
For a richer flavor, use chicken broth instead of vegetable broth.
You can add 1 cup of cooked, shredded chicken breast when combining the tortellini and sauce for added protein.