Toss shallots, vinegar, and sugar in a large bowl; season with salt and pepper. Let sit for 15 minutes or more. Then slice peaches and toss with some of the pickling liquid.
Place burrata on a plate and slightly tear open at the center. Arrange the sliced peaches around the burrata, decorate with basil and sprinkle on the pickled shallots. Drizzle with oil and about 1 tablespoon reserved pickling liquid. Season with salt and pepper and sprinkle on lime zest.
Where Do I Buy Burrata? I used to be able to find burrata cheese only at Whole Foods. But over the past couple of years, my local Kroger chain, King Soopers carries it in the specialty cheese section. It comes in a container labeled 16 ounces, which includes two small balls of burrata. One ball is a perfect amount to make this salad.Can I Make This Ahead of Time? Surprisingly yes. The acid in the vinegar will preserve and keep the peaches from turning a dark color. Just make sure the peaches are tossed well. Arrange the salad on the platter and cover well with plastic wrap. I've made this a few of hours before serving time and it still looks beautiful and fresh.Storing; Got leftovers? Just transfer to an air tight container and store in the refrigerator overnight. The peaches hold up well and the burrata cheese will stay moist and creamy. One day is about the length of time this will keep. If Peaches Are Out of Season, substitute Winter fruits such as pears, apricots or nectarines.