Divide barbecue sauce, ½ for grilling and the other ½ for passing at the table.
Place kebabs on grill, turning one-quarter turn every 2 to 2 ½ minutes until well browned and slightly charred, about 8 minutes for breasts or 10 minutes for thighs. Brush top surface of kebabs with ¼ cup sauce; flip and cook until sauce is brown in spots, about 1 minute. Brush second side with remaining ¼ cup sauce; flip and continue to cook until brown in spots and instant-read thermometer inserted in center of meat registers 160 degrees for breasts and 175 degrees for thighs, about 1 minute longer. Remove kebabs from grill and let rest for 5 minutes. Serve, passing reserved barbecue sauce separately.