Chicken: Open chicken breasts and place butterfly-style on grill over white-hot coals of live oak and pecan wood. Grill 3-5 minutes per side or until cooked through. Remove from grill. When chicken is cool enough to handle, hold breast halves together and stuff each breast with one slice of prosciutto and 1/8 cup Fontina cheese. Secure with wooden picks. Set aside in a warm place.
Lemon Butter and Topping: In a large sauté pan over medium heat, combine clarified butter, garlic, and onion and sauté until tender. Deglaze pan with white wine. Add unsalted butter, salt, and pepper. Add mushrooms to lemon butter and sauté 1 to 2 minutes or until cooked. Add basil and lemon juice and stir to combine. Place chicken on a serving platter and top with mushrooms in lemon-basil butter and serve.