In a microwave-safe bowl, mix sugar, ginger, and 1 tablespoon water. Cut lime leaf crosswise into thin slivers; add to bowl. Heat in a microwave oven on full power (100%) until hot, about 30 seconds. Add lime juice and fish sauce, stir, and let dressing stand.
Rinse cantaloupe and honeydew; cut off and discard peel, cut melons in half, and scoop out and discard seeds. Cut each melon into 1-inch-thick slices and arrange on a platter.
Place shrimp in a colander and rinse under cold water; lay on towels to drain. Scatter shrimp over melon. Stir minced chili (to taste) into dressing, then spoon dressing over melon and shrimp. Sprinkle basil and cilantro over salad, then top with peanuts. To serve, spoon portions, including dressing, onto plates.