Season the chicken with salt and pepper and place in a heavy pot with the onion, garlic and just enough water to cover (about 1-1/2 cups). Add the barbecue sauce, vinegar and a few drops hot sauce and bring to a boil. Reduce the heat and simmer until the chicken is cooked through, about 15 minutes. Remove the chicken from the sauce and shred with two forks.
Boil the sauce, skimming occasionally, until reduced by half, about 15 minutes. Season with salt and pepper. Add the shredded chicken and heat through. Spoon the mixture onto the rolls and top with the cheese.