Brine the chicken: Combine 4 cups water, 2/3 cup salt, and the brown sugar in a medium saucepan. Bring to a boil and whisk until the salt and sugar are completely dissolved. Turn off the heat and transfer to a bowl or pot large enough to submerge the chicken. Stir in 6 cups of ice water and let cool completely. Submerge the chicken in the cooled brine, cover and refrigerate for at least 6 hours or up to overnight.
Take the chicken out of the fridge 30 minutes before cooking. Remove from the brine and pat dry completely.
Prepare a grill for medium-high indirect heat: For gas grills (with 3 or more burners), turn all the burners to medium-high heat; after about 15 minutes turn off one of the middle burners. For charcoal grills, bank one chimney starter-full of lit and ashed-over charcoal briquettes to one side of the grill. Set up a drip pan on the other side to avoid flare-ups. (Be sure to consult the grill manufacturer’s guide for best results.)
Whisk together the paprika, onion powder, dried oregano, dried thyme, cayenne, garlic powder, 1/4 teaspoon ground black pepper and the remaining 1/2 teaspoon salt in a small bowl. Brush the chicken all over with the melted butter, and then pat on all of the dry rub mixture.
Grill the chicken: Place the chicken breast-side up on the indirect side of the grill, with the legs facing the hotter side of the grill. Cover and cook until an instant-read thermometer reads at least 165 degrees F in the thickest part of the thigh (avoiding the bone), 50 to 60 minutes. The chicken will be quite dark and blackened in spots.
Remove from the grill and let rest on a cutting board for at least 20 minutes before carving.
All grills are different and you know yours best. Cooking times may vary, so adjust as needed.