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.