Preheat oven to 350°. Remove giblets and neck, and rinse turkey with cold water. Drain cavity well; pat dry. Stir together molasses, next 5 ingredients, and 1 tablespoon kosher salt in a small bowl.
Starting at neck cavity, loosen turkey skin from breast and drumsticks without totally detaching skin; rub 1/4 cup molasses glaze under skin. Replace skin. Sprinkle kosher salt and black pepper over entire bird. Place shallot and apples in turkey cavity. Tie ends of legs together with kitchen string; tuck wingtips under. Place turkey, breast side up, on a roasting rack in a large, heavy roasting pan.
Roast 1 hour and 30 minutes; brush with reserved molasses mixture, and add broth to pan. Roast, brushing with glaze every 30 minutes, 1 hour 30 minutes, or until a meat thermometer inserted in thickest portion of thigh registers 160°. (Shield with aluminum foil to prevent excess browning, if necessary.)
Let stand 30 minutes before carving, reserving pan drippings for Spiked Gravy.