Pour 1/2 cup kosher salt and the cola into a resealable plastic bag, mix until the salt dissolves.
Add the chicken wings, coat with the brine, squeeze out excess air, and seal the bag. Brine in the refrigerator for 2 hours. While the chicken is brining, stir together the vegetable oil, brown sugar, 2 tablespoons kosher salt, garlic powder, black pepper, mustard powder, and paprika in a small bowl to form a paste; set aside.
Preheat an outdoor grill for medium heat, and lightly oil the grate.
Drain the chicken wings in a colander set in the sink, then pat dry with paper towels.
Place the wings into a large mixing bowl, then mix in the spice paste until evenly coated.
Grill the wings on the preheated grill, turning occasionally, until the chicken is no longer pink at the bone, 10 to 15 minutes.
