Remove giblets and neck from turkey; discard liver. Reserve neck and giblets. Pat dry. Trim excess fat. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat. Lift wing tips up and over back; tuck under turkey.
Combine butter, 2 teaspoons rosemary, 1 teaspoon salt, 1/4 teaspoon pepper, coriander, and garlic in a small bowl. Rub butter mixture under loosened skin and over breasts and drumsticks. Tie legs together with kitchen string. Place remaining 1 teaspoon rosemary, reserved giblets, neck, carrots, celery, and onion in the bottom of a large roasting pan coated with cooking spray. Add 1/2 cup broth. Place roasting rack in pan. Arrange turkey, breast side up, on roasting rack. Bake turkey at 425° for 30 minutes.
Reduce oven temperature to 325° (do not remove turkey from oven). Add 1/2 cup broth to pan. Bake an additional 30 minutes. Rotate pan in oven; bake an additional 30 minutes or until a thermometer inserted into meaty part of thigh registers 165°. Remove from oven; let stand on a cutting board for 30 minutes. Discard skin.
Place a large zip-top plastic bag inside a 4-cup glass measure. Strain pan drippings into bag; let stand 10 minutes. Discard solids. Seal bag; snip off 1 bottom corner of bag. Drain pan drippings into a medium saucepan, stopping before fat layer reaches the opening. Add remaining 2 cups broth to pan; bring to a boil. Combine 3 tablespoons flour and 3 tablespoons water in a small bowl, stirring with a whisk until smooth. Stir the flour mixture into broth mixture, and boil 1 minute or until thickened, stirring constantly. Stir in remaining 1/4 teaspoon salt and remaining 1/4 teaspoon pepper. Serve gravy with turkey.