Stir together apple juice, bourbon, and brown sugar. Stir until the sugar dissolves.
Remove the giblets and neck from the turkey, and rinse the turkey with cold water. Drain the cavity and pat dry. Place the turkey in a large roasting pan. Dip the cheesecloth into the apple juice mixture. Wring dry. Cover the turkey with the cheesecloth. Pour the apple juice mixture over the cheesecloth, coating completely. Cover and chill for 12 to 24 hours, basting occasionally with the marrinade.
Preheat oven to 325 degrees. Remove the turkey from the pan, discarding cheesecloth and reserving 3 cups of the marinade. Sprinkle the cavity with your desired amount of salt; rub into the cavity. Sprinkle your desired amount of salt and pepper on the skin. Rub it into the skin. Tie the legs together with string; tuck the wingtips underneath.
Arrange the celery, carrots, apples, and onion in a single layer on a roasting pan. Place the turkey, breast side up on the celery mixture. Pour the reserved marinade over the turkey in the pan.
Bake at 325 degrees for 3 hours and 15 minutes to 4 hours or until a meat thermometer inserted into the thickest portion of the thigh reads 165 degrees, basting every 30 minutes with the pan juices and shielding with aluminum foil after 2 hours 30 minutes to prevent excessive browning, if necessary. Remove from the oven, and let stand for 30 minutes.
Transfer the turkey to a serving platter. Reserve 2 1/2 cups pan drippings. Pour the drippings through a fine wire mesh stainer into a large measuring cup, discarding solids.
Melt the butter in a saucepan over medium heat. Whisk in the flour and cook, whisking constantly, 1 to 2 minutes or until smooth. Gradually add the drippings, and bring to a boil. Reduce heat to medium, and simmer, stirring the mixture occasionally, 5 minutes or until the gravy thickens. Add up to 1/2 cup chicken broth to achieve the desired consistency. Sir in salt and pepper to taste. Serve the turkey with the warm gravy.