Preheat grill to medium high heat.
Cover 1/2 of each breast section with one slice of prosciutto and one slice Fontina cheese.
Fold chicken breast half flap over the dressed chicken breast half.
Season with pepper and salt.
Place on hot grill. Cook, turning once, until done and internal temperature reaches 165°F as measured by an instant-read thermometer.
In a large sauté pan, place 1 tablespoon butter.
Place pan over medium heat.
Add mushrooms, onions and garlic. Sauté until softened.
Add white wine and lemon juice. Lower heat to medium-low. Simmer approximately 10 minutes to reduce.
A little at a time, add remaining 3 tablespoons butter until butter melts and mixture emulsifies.
Add basil.
Place one chicken breast on each serving plate.
Spoon lemon butter mushroom basil sauce on top of the chicken.
Serve with pasta or mashed potatoes.