Butterfly thickest section of chicken breasts to create 2 lobes.
Pound each breast between 2 sheets of plastic wrap until 1/4 - 1/2-inch thick.
Place flattened chicken breasts on a plate.
Place desired amount of cheese stuffing on one lobe of each chicken breast. Gently press stuffing down so it resembles a hockey puck.
Fold over other lobe of chicken breast - It does not have to seal.
Heat a large sauté pan over medium heat.
Add olive oil. Heat until shimmering.
Place flour in a shallow pan. Season, to taste, with salt and pepper.
Dredge stuffed chicken breasts in flour, shaking off excess.
Sauté chicken breasts with preheated oil, cooking until each side is golden brown.
Remove chicken from pan and place in a baking dish.
Bake, in preheated oven, for 10 - 20 minutes, or until juices run clean and center of chicken reaches 165°F.
Add onions to sauté pan and stir to loosen chicken drippings.
After 2 minutes, add mushrooms. Sauté until onions are translucent.