Preheat grill to medium-high heat and lightly oil the grates.
Brush cut sides of nectarines with olive oil and place on grill, cut side down.
Grill for 3-4 minutes until grill marks form and nectarines are slightly softened. Remove and set aside.
On a serving platter, spread arugula as a base. Layer grilled nectarines and torn prosciutto on top.
Tear burrata into pieces and place over salad.
Drizzle with olive oil and balsamic glaze. Sprinkle fresh basil, salt, and pepper to taste.
Serve immediately while nectarines are warm or at room temperature.