Follow these steps to bring this delightful dish to life:

1. Preheat your grill to medium heat. If you don’t have a grill, a grill pan works just as well.
2. Brush the cut sides of the peaches with olive oil to prevent sticking and enhance their natural sweetness.
3. Place the peaches on the grill, cut side down, and grill for about 4-5 minutes until they have nice grill marks and are tender.
4. Remove peaches from the grill and let them cool slightly. Slice each half into wedges.
5. On a large platter, arrange the grilled peaches, mozzarella slices, and fresh basil leaves in an alternating pattern.
6. Drizzle with remaining olive oil and balsamic glaze, then season with salt and pepper to taste.
7. If using arugula, add a handful around the edges of the platter for an extra layer of flavor.

The key to this recipe is the quality of the ingredients; ripe peaches and fresh mozzarella make all the difference. Remember, cooking is about balance, so feel free to adjust the seasoning to your preference.