Heat oven to 350°. Set a rack over a rimmed baking sheet. Lay prosciutto on rack and bake until crisp, 8 to 10 minutes. Let cool, then break into shards and chips.
Shave off ribbons of cantaloupe onto a large serving platter, using a vegetable peeler, mandoline, or very sharp knife. Sprinkle prosciutto and mint over melon shavings. Drizzle oil over salad.
Note: Nutritional analysis is per 3/4-cup serving.