Stir together kosher salt, next 4 ingredients, and 2 qt. water in a large, deep food-safe container or stockpot until sugar is dissolved. Add turkey. Chill 8 hours or overnight, turning once.
Coat cold cooking grate of grill with cooking spray, and place on grill. Light 1 side of grill, heating to 350° to 400° (medium-high) heat; leave other side unlit. Remove turkey from brine. Rinse turkey, drain well, and pat dry with paper towels.
Stir together table salt and next 5 ingredients. Rub skin of turkey with mixture.
Place turkey, skin side down, over lit side of grill, and grill, without grill lid, 4 to 5 minutes or until slightly charred. Transfer to unlit side, skin side up. Grill, covered with grill lid, 30 to 40 minutes or until a meat thermometer inserted into thickest portion registers 165°. Return turkey, skin side down, to lit side, and grill, covered with grill lid, 4 to 5 minutes or until skin is crisp.
Remove turkey from heat; cover loosely with aluminum foil. Let stand 10 minutes. Serve with salsa verde.
*Frozen turkey breast, thawed, may be substituted.