Place chicken in a large baking dish. Scatter mushrooms around and sprinkle with Swiss cheese. In a small bowl, combine cream of mushroom soup, sour cream, and chicken broth.
Pour sauce mixture over chicken.
Bake the chicken breasts in the preheated oven until no longer pink in the center and the juices run clear, about 50 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Remove from the oven and sprinkle with Parmesan cheese and return to oven.
