Make balsamic reduction by bringing vinegar to a simmer in a small sauce pan over medium-high heat. Once it starts to simmer, reduce heat and continue to simmer for about 10 minutes until thickened and reduced by 1/2. It should be able to coat the back of a spoon. It does get thicker when it cools so watch closely at the end so you don't make it too thick... but if you do, place the pan back on the heat over low and add whisk in a little water to thin it.
Turn grill on high and allow it to preheat.
Pound chicken to an even 3/4 inch thickness and then season with salt and pepper. Grill chicken for about 4 minutes on each side (make sure you are closing the grill lid) until done and juices run clear. Add the goat cheese and cook another minute. Remove from grill and let it rest while you toast the buns for a minute.
Assemble the sandwich with lettuce, tomatoes, onion, and chicken. Drizzle the balsamic reduction and then close the bun. Serve.