Preheat a grill or cast-iron grill pan to medium-high heat.
Lightly brush the cut side of the peach halves with 1 tablespoon of olive oil.
Place peaches cut-side down on the grill and sear for 3 to 4 minutes until distinct grill marks appear and the fruit begins to soften.
Remove peaches from the grill and set aside to cool slightly for 2 minutes.
Arrange the baby arugula as a base layer on a large serving platter.
Place the grilled peach halves on top of the arugula.
Carefully tear or break the burrata balls into large chunks and distribute them among the peaches.
Drizzle the salad with the remaining tablespoon of olive oil and the balsamic glaze.
Top with toasted pine nuts and fresh basil chiffonade.
Season with flaky sea salt and freshly cracked black pepper immediately before serving.