Mix salt, pepper, granulated sugar, and paprika in a small bowl. Arrange turkey on a cutting board breast side up.
Remove legs (keeping thigh and drumstick intact) by slicing through the skin that joins breast and leg, then bend leg down to expose joint and cut completely through the joint.
Try to leave as much skin on breast as possible. Place legs on a large rimmed baking sheet. Set aside neck and giblets in cavity for making gravy.
Arrange turkey on baking sheet with legs and generously sprinkle legs and turkey all over with dry brine, patting to adhere.
Chill, uncovered, at least 12 hours and up to 2 days.