Whisk together first 5 ingredients in a small bowl. Trim excess fat from turkey. Remove giblets; set aside for another use. Loosen skin from breast by inserting fingers and gently pushing between skin and meat. Rub salt mixture evenly under and over skin. Place turkey on a pan and refrigerate, uncovered, 8 hours or overnight.
Soak wood chips in water 1 hour; drain well. Meanwhile, remove turkey from refrigerator; let stand at room temperature 1 hour.
Light one side of grill, heating to medium heat (350°); leave other side unlit. Place 1 cup wood chips in an aluminum foil packet, and place over charcoal or heating element. Place a disposable aluminum foil pan under grill rack on unlit side. Pour 2 cups water in pan. Coat grill rack with cooking spray, and place turkey on unlit side of grill.
Grill turkey, covered with grill lid, 2 hours and 30 minutes or until a thermometer inserted into the thickest part of the breast registers 165°, turning turkey and adding remaining 1 cup wood chips halfway through cooking time. Remove turkey from grill; let stand 30 minutes before slicing.