To prepare brine, combine first 6 ingredients in a large stockpot, stirring until salt dissolves.
To prepare turkey, remove and reserve giblets and neck from turkey. Rinse turkey with cold water; pat dry. Trim excess fat. Add turkey to pot, turning to coat. Cover and refrigerate 24 hours, turning occasionally.
Preheat oven to 375°.
Bring cola and 1/2 cup syrup to a boil in a small saucepan; cook 1 minute.
Combine thyme, sage, seasoning, and 1/2 teaspoon pepper. Remove turkey from brine, discarding brine; pat dry. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat. Rub thyme mixture under loosened skin; sprinkle inside body cavity. Place 4 garlic cloves and onions in body cavity. Tie ends of legs together with twine. Lift wing tips up and over back; tuck under turkey.
Place turkey on a broiler pan coated with cooking spray. Insert a meat thermometer into meaty part of a thigh, making sure not to touch bone. Bake at 375° for 45 minutes. Pour cola mixture over turkey; cover with foil. Bake an additional 1 hour and 45 minutes or until thermometer registers 180°. Remove turkey from pan, reserving drippings for gravy. Place turkey on a platter. Cover loosely with foil; let stand 10 minutes. Remove twine. Discard skin.
To prepare gravy, while turkey bakes, combine reserved giblet and neck and the broth in a saucepan; bring to a boil. Cover, reduce heat, and simmer 45 minutes. Strain mixture through a colander into a bowl, discarding solids.
Place a zip-top plastic bag inside a 2-cup glass measure. Pour pan drippings into bag; let stand 10 minutes (fat will rise to the top).
Seal bag; carefully snip off 1 bottom corner of bag. Drain drippings into broiler pan, stopping before fat layer reaches opening; discard fat. Add broth mixture. Place broiler pan on stovetop over medium heat, scraping pan to loosen browned bits. Combine milk and cornstarch in a small bowl, stirring well with a whisk; add to pan. Bring to a boil; cook 1 minute, stirring constantly.
Strain gravy through a sieve into a bowl. Discard solids. Stir in 1/4 teaspoon salt and 1/4 teaspoon pepper.