1. Preheat oven to 450F. Stir together sugar and ginger in a small bowl. Arrange papaya halves, cut sides up, in a 10-by-13-inch baking dish. Sprinkle sugar mixture evenly over halves.
2. Bake, brushing papaya edges with melted sugar mixture (it will collect in well of fruit) 2 or 3 times, until mixture is bubbling and papaya edges are beginning to darken, 35 to 40 minutes.
3. Sprinkle each serving with a pinch of cayenne. Serve with lime wedges.