Preheat a grill or grill pan to medium-high heat.
Brush the cut sides of the peach halves lightly with olive oil.
Place peaches cut-side down on the grill for 3 minutes until grill marks form. Flip and grill 2-3 minutes more.
Once cool enough to handle, slice each peach half into 3-4 wedges.
Scatter arugula across a large serving platter.
Arrange the grilled peach wedges over the arugula. Tear the burrata into large pieces and nestle among the peaches.
Drizzle with balsamic glaze and olive oil. Scatter torn basil, and finish with flaky salt and black pepper.