Add chicken breast to a small saucepan. Add brown ale and balsamic vinegar. Bring to a simmer and cover, simmering over med-low heat for 20 minutes, flipping once. Beer and balsamic vinegar should turn into a syrup around chicken. If still watery, allow sauce to thicken while you shred the chicken. Remove chicken from pan and use two forks to shred. Return to the pan with sauce and turn off the heat.