Heat gas grill to medium-high heat. Alternatively, heat a grill pan over medium-high on top of the stove.
Slice the peaches in half and remove the pit. Slice each piece in half again lengthwise. Brush both sides of each slice lightly with 1 tablespoon olive oil. Grill peach slices 2 to 3 minutes per side. You want them to slightly caramelize without them getting too soft. Let them cool slightly before adding them to the platter.
Slice the tomatoes into rounds, about 1/4" thick.
Alternate peach and tomato slices on a platter. Break the burrata over the top of the tomatoes and peaches. Tear basil leaves into small pieces and sprinkle all over. Drizzle remaining 1/2 tablespoon olive oil and aged balsamic vinegar over peaches and tomatoes.
Serve within an hour of dressing.
Notes
* Instead of burrata you can use mozzarella. Feel free to double the amount as you can never have enough cheese. * Aged balsamic vinegar is a bit sweeter and thicker than regular balsamic vinegar. You can also use regular balsamic vinegar or balsamic syrup.