Heat coals or gas grill.
Pound chicken with salt to 1/8-inch thickness between 2 pieces of waxed paper.
Mix sugar and cinnamon. Coat apple slices with sugar mixture. Divide apple slices among chicken breast halves. Fold chicken around apples; secure with toothpicks.
Cover and grill chicken 4 to 6 inches from medium heat 20 to 25 minutes, turning once, until juice is no longer pink when centers of thickest pieces are cut. Remove toothpicks.
Mix apple cider, sage and cornstarch in saucepan. Cook over medium heat, stirring constantly, until thickened and bubbly. Spoon over chicken.