Preheat the oven to 350 degrees F. Place the prosciutto slices on a foil-lined baking sheet. Place in the oven and bake until firm, golden around the edges, and almost crisp, about 18 minutes. Set aside and let cool.
Place half the melon in the blender with half the tomatoes. Pulse the blender until the mixture is pureed. Pour the soup in a large bowl. Puree the remaining melon and tomatoes and add to the bowl. Add the salt and pepper.
Ladle the soup into bowls. Crumble the cooled prosciutto crisps over the top of the soup. Sprinkle with basil and serve.