Drizzle peaches with balsamic vinegar and then sprinkle liberally with brown sugar. (I like to do this about an hour before grilling so the flavors soak in, but they’ll be delicious either way.)
Brush grill with a dab of oil. Place peach halves face-down on grill. Turn after 3 to 4 minutes, after they’ve given up the brightness in their color. Sprinkle flat side (now facing up) 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.
Garnish with basil on and around peaches. Serve with ice cream or by themselves.