Season chicken w/ salt and pepper.
Heat 2 T oil in large skillet over medium heat. Cook chicken 2-3 minutes per side, until golden brown. Remove from heat. (chicken does not need to be cooked through at this point)
Add onions. Cook until mostly softened, about 3 minutes, stirring often. Add garlic and cook 1 minute more.
Add broth and balsamic. Using a rubber spatula, scrape up all the yummy bits from the bottom of the pan. Bring to a bubble and allow mixture to reduce by half.
Add chicken (and the juices that may have rendered out) back to the pan. Cook until chicken is no longer pink in the middle. Turn several times to coat in sauce.
Add 1T butter to pan to finish the sauce. Top with chopped parsley.