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1
Mix paprika, savory, cumin, mustard powder, cayenne, and 1/4 cup brown sugar in a small bowl. Transfer 3 Tbsp. spice mixture to a medium bowl; mix in salt and remaining 3 Tbsp. brown sugar. Set remaining mixture aside for the next day. Cover turkey with dry brine, packing on until you've used it all. Chill on a rimmed baking sheet uncovered 8-12 hours.
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2
Thoroughly rinse turkey to remove brine; pat dry. Rub with reserved spice mixture and let sit 2 hours to bring to room temperature.
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3
Prepare grill for medium-high, indirect heat (for a charcoal grill, bank coals on one side of grill; for a gas grill, leave one or two burners off) and lightly oil grates. If using wood chips: Scatter over coals, if using a charcoal grill; place in a disposable foil pan and set over lit burner, if using a gas grill.
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4
Place turkey, skin side up, over indirect heat, breast halves facing each other, with legs closest to direct heat. Place oven thermometer in center of indirect heat zone; cover. Grill 20 minutes, then rotate halves so neck end is closest to coals. Check temperature inside grill; thermometer should register 325u00b0F (open or close vents as needed). Continue to grill turkey, checking for doneness after 20 minutes, then again every 10 minutes, until an instant-read thermometer inserted into thickest part of breast registers 150u00b0F, about 1 hour total.
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5
If desired, carefully move turkey over direct heat and turn skin side down. Grill to lightly char skin, about 2 minutes. Transfer to a platter and let rest at least 30 minutes before carving.
