In a medium bowl, combine flour, salt and pepper. Coat chicken well with seasoned flour. In large heavy skillet, heat olive oil and butter over medium-high heat. Add chicken breasts to skillet and saute for 2 minutes on each side, just until golden. Remove chicken to casserole dish, arranging in single layer.
Add onion, garlic and gingerroot to the skillet; cook for 1 minute. Stir in apple slices. Blend in cider, brown sugar, and vinegar. Simmer on low heat 4 to 5 minute. Pour mixture over chicken and bake about 20 minutes. Remove from oven, serving at once. (If sauce does not thicken into glaze; stir 1 teaspoon flour mixed with 3 tablespoons cider. Cook and stir over medium heat, just until thickened. Spoon over chicken)