In a small bowl, combine brown sugar and cinnamon. Cut peaches along the seam all the way around and twist halves off the pit. Brush cut sides with vegetable oil. Cook, cut side down, on a hot grill until fruit has grill marks, 3 to 4 minutes.
Brush tops with oil, turn over, and move to indirect heat. Sprinkle cut sides with cinnamon sugar. Cover grill and cook until sugar is melted and fruit is tender, 10 to 15 minutes. Serve with vanilla ice cream (sprinkle on extra cinnamon sugar if you like).