Pat the turkey breast dry with paper towels, then mix together the salt, pepper, garlic powder, smoked paprika, and brown sugar in a bowl.
Thoroughly rub the turkey with the dry spice mixture, making sure to cover it evenly.
Preheat the smoker or grill to 230°F (110°C), and prepare charcoal or smoking wood for an intense smoky flavor.
Place the turkey on the smoker rack, and smoke for 3-4 hours, or until the internal temperature reaches 165°F (74°C).
Mix together the Worcestershire sauce, apple cider vinegar, and BBQ sauce, then brush the turkey with it during the last 30 minutes of smoking.
Once cooked, remove the turkey from the smoker, and let it rest for 15 minutes to allow the juices to redistribute.
Thinly slice and serve fresh on buns with grilled vegetables or coleslaw.
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