Preheat grill to high.
Mix dressing, whisk, and set aside.
Place peaches cut-side down on oiled grates. Grill uncovered until well charred (about 4 minutes), turning to grill on all sides.
Toss arugula and hazelnuts with dressing. Put on plate.
Place peaches and mozzarella on salad. Drizzle with balsamic glaze.