Preheat oven to 450°F. Stir together sugar and ginger. Arrange papaya halves, cut side up, in a 10-by-13-inch baking dish. Sprinkle sugar mixture evenly over halves.
Bake, brushing papaya edges with melted sugar mixture (it will collect in well of fruit) two or three times, until mixture is bubbling and papaya edges are beginning to darken, 35 to 40 minutes.
Remove from oven, and sprinkle fruit with cayenne. Serve immediately with lime wedges.
2 tablespoons dark brown sugar
1/4 teaspoon ground ginger
2 papayas (Solo; 14 ounces each), halved lengthwise and seeded
1/4 teaspoon cayenne pepper
1 lime, cut into 4 wedges, for serving