Heat oven to 400 F. Arrange the prosciutto in a single layer on a baking sheet.
Bake until crisp, 8 to 10 minutes. Trim and discard the rind from the watermelon.
Cut the flesh into 1/4-inch-thick triangles.
Place the watermelon in serving dishes and sprinkle with the radishes, scallions, mint, and peanuts.
Drizzle with the lime juice and oil. Season with 1/4 teaspoon salt and 1/8 teaspoon pepper. Break the prosciutto into pieces and sprinkle over the salad before serving.