Slice the bread into 3/4" slices. Gently oil or butter both sides of the bread.
Heat grill or grill pan over medium-high heat. Place bread slices on the hot grill and cook until grill marks form and the bread slices are lightly toasted, about 2 to 3 minutes. Using tongs or a spatula, turn the slices over and cook them on the other side until grill marks form, another 2 to 3 minutes. Season with sea salt. Set the bread aside while you grill the peaches.
To grill the peaches. Preheat your grill or grill pan to medium-high. Brush the cut side of the peach halves with oil and place on the grill for 4 minutes or until grill marks appear and the fruit starts to caramelize. Use a pair of tongs to remove the peaches from the grill. Let the peaches cool for a few minutes or until they are cool enough to handle. Slice the grilled peaches.
When ready to serve, place the burrata on a platter or serving dish. Place the sliced peaches around the burrata. Drizzle honey over the burrata and grilled peaches and season with a little sea salt. Garnish with chopped pistachios and fresh mint. Arrange the bread slices around the cheese and peaches and serve.