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 burrata on a plate, roughly torn and pulled apart at the center. Top with sliced peaches, basil and pickles shallots.
Drizzle with oil and 1-2 tablespoons reserved pickling liquid.
Season with salt and pepper and finish with a zesting of lime.