Build a charcoal fire in an outdoor grill and let burn until the coals are covered with white ash. Let the fire burn down to medium-hot (you should be able to hold your hand at grate level for 3 seconds). In a gas grill, preheat to Medium.

Tear off six 12-inch squares of aluminum foil. Fold one foil square in half. On the bottom half, place 2 tablespoons brown sugar, 2 tablespoons Kahlua, 2 tablespoons butter, and 1 banana. Sprinkle with a pinch of cinnamon. Fold the foil over to form a packet, crimping the open edges to seal. Repeat with the remaining ingredients. The packets can be prepared up to 1 hour ahead and kept at room temperature.

Grill the packets until the sugar, Kahlua, and butter are boiling, 5 to 7 minutes. Open the packets, pour over bowls of ice cream and serve immediately.