In a medium mixing bowl combine the ingredients for the rub and mix well. Add the Turkey breast, and fully coat it in the rub. Cover the bowl and put it in the fridge for two hours to overnight.
Turn on half your grill and heat it to 400°F. Put the turkey breast on the half of the grill that is off, skin side up, close the lid and let it cook for 45 minutes.
While the turkey is on the grill, make the sauce by whisking the sauce ingredients together in a medium mixing bowl. Divide the sauce in two. Set half the sauce aside and use half to brush on the turkey.
Once the turkey breast has been cooking for 45 minutes, start brushing it with the sauce every ten minutes for another 30 to 40 minutes or until the internal temperature of the turkey reaches a temperature between 165°F and 170°F. Flip the turkey breast every time you brush it.
To make the coleslaw, mix the mayo, cider vinegar, kosher salt, and sugar then add the cabbage and apple. Mix until all the cabbage and apple are coated in the sauce.
Once the turkey is cooked, take it off the grill and allow it cool for 10 to 15 minutes at room temperature. Using your hands pull the meat off the bone and pull it into small pieces. Put the pulled turkey in the second half of the sauce and mix well.
To assemble the sandwich, toast four buns and brush with some of the mustard-cider barbecue sauce. Put one quarter of the turkey on each bun and top each with one quarter of the coleslaw.
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