1. Preheat grill. Drizzle peaches with balsamic vinegar and then sprinkle liberally with brown sugar.
2. Brush grill with a dab of oil. Place peach halves face-down on grill, 3 to 4 minutes.
3. Peaches should be less bright in color. Flip and sprinkle with more brown sugar. Remove after another couple of minutes, when they're soft all the way through when you poke them with a fork.
4. Garnish with basil on and around peaches. Serve with ice cream or by themselves.