Combine peaches and bourbon in a bowl. Let soak for 30 minutes.
Preheat barbecue on medium heat and lightly oil the grill plate. Grill bread slices until grill marks appear, about 2 minutes per side. Transfer to a plate.
Grill peaches until dark grill marks form, 2 to 3 minutes. Transfer to a bowl.
Place Burrata cheese and rocket on a platter. Drizzle olive oil on top; season with salt and pepper. Add grilled peaches to the platter. Drizzle balsamic glaze over Burrata cheese, rocket and peaches. Scatter prosciutto on top. Serve with grilled bread.