Preheat a grill or grill pan to medium-high heat.
Lightly brush the cut sides of the nectarines with 1/2 tablespoon of olive oil.
Place nectarines cut-side down on the grill and cook for 3 to 5 minutes until distinct grill marks appear and fruit slightly softens.
Arrange the arugula on a large serving platter.
Place the grilled nectarines on top of the arugula bed.
Carefully tear the burrata balls into large pieces and nestle them among the nectarines.
Drape slices of prosciutto around the cheese and fruit.
Drizzle the remaining olive oil and the balsamic glaze over the entire salad.
Season with flaky sea salt and black pepper, then garnish with torn basil leaves and serve immediately.