Preheat grill or grill pan to medium. In a small bowl, mix sugar and cinnamon. Brush cut side of peaches lightly with butter.
Oil grill. Place peaches cut-side down on grill and cook until grill marks appear, 3 to 4 minutes. Brush bottoms lightly with remaining butter and turn peaches over. Sprinkle sugar mixture over cut side of peaches, cover grill and cook until sugar has melted and caramelized and peaches are easily pierced with a sharp paring knife, 3 to 6 minutes longer. Serve immediately, with ice cream, if desired.