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Heat grill to medium-high. Remove turkey from the brining solution, rinse well under cold water and pat dry with paper towels. Rub the skin side of the turkey with the rub. Using tongs, dip paper towels into the oil and oil the grates of the grill. Place the turkey on the grill, rub side down and grill until slightly charred and a crust has formed, about 4 to 5 minutes. Reduce the heat of the grill to medium, turn the turkey over and continue cooking, with the lid closed for 30 to 40 minutes or until an instant-read thermometer inserted into the center of the breast registers 155 degrees F (the temperature will continue to rise off the heat). Remove from the heat, loosely tent with foil and let rest 10 minutes before slicing. Slice into 1/2-inch thick slices.
While turkey is resting, brush the cut side of oranges with oil and place cut side down on the grill and grill until slightly charred. Ladle sour orange sauce onto a platter and top with the sliced turkey. Garnish with grilled oranges and parsley leaves.
Strain the sauce into a bowl, whisk in the butter, and season with salt and pepper. Stir in the thyme and parsley and serve hot.