Cook the pasta according to package directions until al dente. Reserve 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 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 aside.
Add the butter to the same skillet. Add the sliced mushrooms and 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 low.
Stir in the grated Parmesan cheese until the sauce is smooth and thickened. If the sauce is too thick, add a splash of the reserved pasta water until you reach your desired consistency.
Return the cooked chicken to the skillet. Add the drained pasta to the sauce and toss everything together until the pasta is evenly coated. Add more pasta water if needed to loosen the sauce.
Serve immediately, garnished with fresh parsley and extra Parmesan cheese.
Notes
For a thicker sauce, let the cream simmer for an extra minute or two before adding the cheese.
You can substitute half-and-half for heavy cream, but the sauce will be slightly less rich.
Add 2 cups of fresh spinach during the last minute of cooking the sauce for added greens.