Cook the pasta according to package directions until al dente. Reserve about 1 cup of the starchy 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 to 7 minutes. Remove the chicken from the skillet and set it aside.
Add the sliced mushrooms to the same skillet. Cook until they release their moisture and start to brown, about 5 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 medium-low.
Stir in the grated Parmesan cheese until the sauce is smooth and thickened slightly. Taste the sauce and add more salt and pepper if needed.
Return the cooked chicken to the skillet. Add the drained pasta to the sauce. Toss everything together to coat the pasta evenly.
If the sauce seems too thick, add the reserved pasta water, one splash at a time, until you reach your desired consistency.
Serve immediately, garnished with fresh parsley and extra Parmesan cheese.
Notes
For a deeper mushroom flavor, use a mix of wild mushrooms instead of just cremini.
If you prefer a lighter sauce, substitute half of the heavy cream with whole milk, but the sauce will be less rich.
You can add a pinch of nutmeg to the sauce for added depth.