Toss shallots, vinegar, and sugar in a large bowl; season with salt and pepper. Let sit for 15 minutes then pour off pickling liquid, reserving liquid.
Place the burrata on a plate, roughly torn and pulled apaprt at the center. Top with sliced peaches, basil and pickled shallots.
Drizzle with oil and 1-2 tablespoons of the reserved picking liquid.
Season with salt and pepper, and finish with a zesting of lime.