Remove giblets and neck from turkey; discard, if desired. Rinse turkey thoroughly with cold water; pat dry. Tie ends of legs to tail with cord, or tuck flap of skin around tail. Lift wing tips up and over back, and tuck under bird. Stuff cavities of turkey with quartered apple, onion, and parsley sprigs. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat.
Combine apple butter, sugar, mustard, and nutmeg in a small bowl; stir well. Spread apple butter mixture under loosened skin and rub into the body cavity. Sprinkle bird with 1 teaspoon salt, 1 teaspoon black pepper, and ground red pepper.
Pour broth and enough water to equal 3 cups liquid into the bottom of the pan. Place turkey on a rack coated with cooking spray; place rack in a shallow roasting pan. Insert meat thermometer into meaty part of thigh, making sure not to touch bone. Bake at 325° for 3 hours or until thermometer reaches 160°. Cover turkey loosely with foil; let stand 10 minutes. Discard skin.
Remove turkey from pan, reserving 1 1/2 cups cooking liquid; reserve remaining cooking liquid for another use. Place turkey on a platter. Set aside; keep warm. Pour reserved 1 1/2 cups cooking liquid into a zip-top plastic bag. Snip off 1 corner of bag; drain liquid into measuring cup, reserving 1 cup of broth, and stopping before the fat layer reaches the opening. Drain fat layer into another bowl, reserving 2 tablespoons. Discard remaining fat.
Combine reserved fat and flour in a medium saucepan over medium heat, and stir until smooth. Stir in reserved 1 cup broth, cider, and milk, and bring to a boil, reduce heat, and simmer, uncovered 3 minutes, stirring constantly until thick and bubbly. Stir in remaining salt and black pepper. Serve gravy with skinned turkey.