Prepare and heat charcoal or gas grill. (Or heat grill pan to medium high heat).
Cut around each plum completely to the pit, all the way around. Twist to split in half. Cut out the pit remaining in the pitted halves.
Grill each plum half for 2 to 3 minutes until dark grill marks appear.
Add plums, and all ingredients to a food processor or blender of choice, and blend until completely smooth.
Lay four chicken breasts on cutting board and drizzle with olive oil, and salt and pepper each side.
Lightly baste chicken with plum sauce and place on hot grill. Cook about 10 minutes each side basting occasionally.
Cook until internal temperature reaches 165 degrees, and no longer as this will result in dry chicken.
Remove from grill, and let rest for 5 minutes before cutting.
Drizzle with additional balsamic vinegar.