Heat oil in medium-size nonstick skillet over medium heat. Add onion; cook until very tender, about 8 minutes. Add apple, 1/2 teaspoon thyme, 1/4 teaspoon salt, 1/4 teaspoon pepper and 1/4 cup cider; cook until apples are tender, about 5 minutes. Remove from heat. Let cool slightly. Stir in Brie. Divide stuffing into 4 equal portions.
Heat oven to 400°F.
Run fingers under breast skin to separate from flesh. Put one-fourth of stuffing under skin of each breast; press gently to distribute filling evenly. Season chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper. Place in 13 x 9 x 2-inch baking dish.
Bake in 400°F oven until no longer pink near bone, 35 minutes (180°F on instant-read meat thermometer). Remove chicken to platter; keep warm.
Skim fat from baking dish. Scrape drippings into small saucepan. Heat over medium heat. Add remaining 1/2 cup apple cider; cook to reduce by half. Stir in remaining 1/2 teaspoon thyme and 1/4 teaspoon salt. Spoon over chicken.