Preheat the oven to 400°F. Arrange the bread slices on a large rimmed baking sheet and brush the tops with the olive oil. Bake for 10 to 12 minutes, until lightly toasted. Remove from the oven and, while the bread slices are still warm, rub the garlic over the tops.
Stir together the peaches, shallot, brown sugar, vinegar, salt, and cinnamon in a small bowl to combine.
To assemble, spread the burrata evenly over the crostini. Spoon the peach chutney over the tops. Garnish each crostini with a basil leaf and drizzle with some balsamic vinegar.