Place chicken in a shallow 2-quart microwave-safe dish with a lid; season with salt and pepper. Cover, and microwave on high until chicken is cooked through, 12 to 16 minutes, turning over halfway through. Drain liquid from dish. Brush chicken with barbecue sauce.
While chicken is cooking, make coleslaw: In a large bowl, whisk together mayonnaise, vinegar, sugar, celery seed, and garlic. Add cabbage and carrot to dressing mixture, and toss to combine. Serve chicken with coleslaw.