Cut the chicken into serving pieces: the breast into halves, the legs and thighs separated, plus the wings, for a total of 8 pieces. Trim away any visible fat and discard. Rinse the chicken well under cold running water. Pat dry with paper towels. Sprinkle the chicken with the salt and black pepper. Place the chicken into a plastic or brown bag. Pour the flour into the bag and shake well to cover the chicken all over. Remove the chicken from the bag and dust off excess flour. Save 1 tablespoon of the flour to use to thicken the gravy. Dice the celery; slice the onions; and mince the garlic. Crush the thyme. Set aside. Heat the oil in a large skillet that has a cover. Add the chicken and saute over mediumlow heat for about 20 minutes, turning occasionally to brown evenly. When golden brown, push the chicken to the side of the pan. Add the reserved tablespoon of flour and saute, stirring until browned. Add the celery, onion, and garlic to the skillet and saute for 5 minutes, stirring occasionally, or until the vegetables are tender. Stir together the chicken and vegetables. Add the chicken broth. Bring to a gentle boil. Reduce the heat to very low, cover, and simmer the chicken for 20 to 25 minutes, or until the chicken is fork-tender and the juices run clear when pierced.