In large bowl, stir together ginger ale, thyme, oil, lime juice, soy sauce, garlic, salt, pepper, cinnamon and allspice. Add chicken; turn to coat. Cover and marinate in fridge 1 to 4 hr.
Cook chicken on grill preheated to medium, 20 to 25 min., or until thickest part of meat reaches 165°F (74°C) on an instant-read thermometer. Turn chicken pieces often. In last 2 min. of grilling, brush with Spirited Mickie Dark & Stormy Rum-Spice BBQ sauce
Alternatively, bake chicken on parchment paper-lined baking sheets in preheated 400˚F (200˚C) oven for 20 min. Brush with barbecue sauce and bake another 8 to 10 min., or until thickest part of meat reaches 165°F (74°C) on an instant-read thermometer.
Shred any leftover chicken, stirring in more barbecue sauce, to serve in buns.