Lightly brush grill with olive oil, then preheat on medium flame. Brush both sides of peach slices with olive oil.
Place peach slices on grill, grilling each side for 1½-2 minutes, until grill marks appear. Place on paper towel-lined plate to cool.
Brush baguette slices with olive oil on both sides. Grill each side for 2 minutes. Remove and set aside to cool.
Place water and blackberries in a small saucepan set over high heat. Bring to a boil, macerating berries with a potato masher or back of a large spoon. Reduce heat to medium and cook for 7-10 minutes.
Set a fine mesh strainer over a small bowl. Pour berry sauce through strainer, pressing the solids to extract all the juice. Discard solids and return juice to saucepan.
Place saucepan back on heat and add balsamic vinegar. Allow to boil 5 minutes, then reduce heat and simmer for 10 minutes until sauce has reduced and thickened. Cool and pour into a small bottle or jar.
Arrange peach slices and basil leaves alternately in a semi-circle on a plate. Place a ball of burrata in the middle, then drizzle with blackberry balsamic. Serve with grilled baguette slices. Enjoy!