In a saucepan, boil water, salt it, then cook the pasta according to the package instructions. When the pasta is cooked, reserve ½ cup of the pasta water and drain the pasta.
Meanwhile, in a skillet, heat oil and add the chicken. Season with salt and pepper and cook the chicken for 4-5 minutes. Use an instant read thermometer to cook the chicken to an internal temperature of 165°F/74°C. Remove the fully cooked chicken onto a plate and set aside.
In the same skillet, melt the butter. Add mushrooms, and cook for 3-4 minutes to sweat them until they soften. Add the garlic and cook for 1 minute.
Add the cream and the parmesan, and whisk or stir until you get a smooth sauce. Add up to half of the pasta water if needed to loosen the sauce and whisk again.
Add the pasta and return the chicken to the skillet, and gently mix everything together until well combined. Taste the pasta with the sauce, and adjust the seasonings to your preference. If needed, more pasta water can also be mixed in to make the pasta saucier.
Garnish with extra parmesan cheese and fresh chopped parsley before serving.