Cook the pasta according to package directions until al dente. Reserve 1/2 cup of the pasta water before draining. Set the drained pasta aside.
Season the chicken pieces with salt and pepper.
Heat the olive oil and butter in a large skillet over medium-high heat. Add the chicken and cook until browned and cooked through, about 5-7 minutes. Remove the chicken from the skillet and set aside.
Add the sliced mushrooms to the same skillet. Cook until they release their moisture and start to brown, about 6-8 minutes.
Add the minced garlic and dried thyme to the mushrooms. Cook for 1 minute until fragrant.
Pour in the heavy cream and chicken broth. Bring the mixture to a gentle simmer, stirring occasionally. Reduce the heat to low.
Stir in the grated Parmesan cheese until the sauce is smooth and thickened. Taste and adjust salt and pepper if needed.
Return the cooked chicken to the skillet. Add the drained pasta to the sauce. Toss everything together until the pasta is fully coated. If the sauce is too thick, add a splash of the reserved pasta water until you reach your desired consistency.
Serve immediately topped with extra Parmesan cheese and fresh parsley.
Notes
For a deeper mushroom flavor, use a mix of wild mushrooms instead of just cremini.
You can substitute chicken thighs for breasts if you prefer darker meat.
If you want a tangier sauce, add 1 tablespoon of Dijon mustard when you add the cream.