In a large zip-top bag, combine chicken, hot sauce, oil and salt and pepper. Shake to coat and refrigerate for 30 minutes, or overnight for stronger flavor.
Heat a medium skillet on medium-high. Add chicken and any hot sauce mixture from bag and cook, turning once, until browned, about 8 minutes. Reduce heat to medium and add broth. Cover and cook for 5 minutes. Remove chicken from sillet; use 2 forks to finely shred chicken. Return to pot and cook uncovered for 10 more minutes, until sauce thickens.
Toast buns and divide chicken mixture, lettuce and carrot among halves. Top with cheese, dividing evenly.