Heat a grill or grill pan to medium-high.
Slice the peaches in half, remove the pits, then slice them in half again to quarter. Lightly brush the cut sides of the peaches with 1 tablespoon of olive oil.
Place the peaches cut side down on the grill and cook, uncovered, until nice grill marks have formed, about 3 minutes per side.
Lay the prosciutto slices on the hot grill or grill pan and cook until crispy, 1 to 2 minutes per side.
To assemble the salad: In a large serving bowl, combine the grilled peaches, arugula, avocado, cantaloupe and basil. Drizzle the aged balsamic vinegar and remaining 2 tablespoons of olive oil on top. Toss to combine.
Crumble the crispy prosciutto on top of the salad and serve immediately.