1. Prepare 1 1/4 cups of stuffing mix according to package directions. Reserve remaining 1/4 cup dry stuffing mix.
2. Preheat oven to 400°F. Grease a shallow 2-quart baking dish with 1/2 table-spoon butter. Place chicken, 1 piece at a time, in a plastic bag or between 2 sheets of plastic wrap. Use a flat mallet to pound chicken to an even 1/2-inch thickness. Divide prepared stuffing into 6 portions and form each into a ball. Set a ball in center of each flattened chicken breast. Roil chicken around stuffing to enclose completely.
3. Place chicken, smooth side up, in baking dish. Sprinkle dry stuffing over chicken and dot with remaining butter. Pour broth around chicken.
4. Bake until chicken is white throughout and stuffing crumbs are browned, about 30 minutes. Serve with pan juices spooned over.