Mix together equal parts vinegar, oil, brown sugar, garlic, and pepper in a medium bowl. Make sure to reserve about three tablespoons of the marinade.
Pour the marinade over the chicken breasts and seal the bag (or container) with a lid. Refrigerate chicken for 30 minutes to 24 hours after coating well.
The grill should be heated to medium-high. You can spray nonstick grilling spray or olive oil on grill grates. (In case of spraying, please turn the grill off, spray, and then turn the grill back on.) Over indirect heat, grill the chicken approximately six minutes (per side), then turn it over and brush with balsamic marinade. Cook for an additional six minutes.
Transfer the chicken to a serving plate as soon as it comes off of the grill.