Grill peaches cut side down on preheated, oiled grill at 450 - 550 degrees F. Grill until well-charred, moving frequently - about 4 minutes.
Toss together arugula, mozzarella, mint, and pistachios (if desired) in a large bowl.
Whisk together olive oil, vinegar, lemon juice, mustard, honey or agave, salt, and pepper in a medium bowl until well combined. Drizzle vinaigrette over arugula; toss to combine. Serve grilled peaches over salad.