Preheat a gas grill to medium heat. If you're using charcoal, grill the peaches after all the other ingredients have been cooked.
Using a sharp knife, carefully slice each peach along the seam to create two halves. Discard the pit and lightly brush the exposed flesh of each peach with olive oil.
Place the peach halves cut-side down on the grill and cook for 4 to 5 minutes, without moving them, until they develop distinct grill marks.
Turn the peaches over and continue to grill on the skin side for an additional 4 to 5 minutes, until the skin is lightly charred and the peaches feel tender.
Take the peaches off the grill. Serve the halves as individual servings with ice cream for dessert, or allow them to cool, slice, and layer on top of a toasted baguette with burrata and a hot honey drizzle.