Pour 1/2 teaspoon oil in a large frying pan; heat over medium heat. Sauté 1/2 cup onion in oil for 4 minutes. Add sautéed onion, prosciutto, cheese, breadcrumbs, and 1 teaspoon rosemary to a bowl; reserve.
Slice a horizontal cut through the thickest portion of each breast to form a pouch. Spoon about 3 tablespoons of breadcrumb mixture into each pouch.
Pour 1/2 teaspoon oil in a large frying pan; heat over medium-high heat. Cook chicken in oil 6 minutes per side or until fully cooked. Take chicken out of pan using a slotted spoon; reserve in a warm place.
In same skillet, sauté 1/4 cup onion for 3 minutes. Stir in 1/2 teaspoon rosemary, broth, wine, and garlic and let boil. Cook for 5 minutes or until liquid is reduced to 3/4 cup. Place chicken back in pan, place lid on pan and let simmer for 2 minutes or until fully heated. Spoon sauce over chicken and top with rosemary sprigs.