Using an electric mixer, beat the ice cream, chopped peaches, gingersnaps and coconut on low speed until just combined. Freeze until firm, 2 to 3 hours.
Preheat a grill to medium. Brush the cut sides of the peach halves with the oil and grill cut side down until golden-brown, about 2 minutes. Turn the peaches over and grill for 1 minute more. Divide among 4 bowls; top each with a scoop of ginger-peach ice cream.