Toss nectarines, onion, vinegar, and sugar in a large bowl; season with salt and pepper. Let sit 10 minutes.
Pour off pickling liquid from nectarines and onion, reserving liquid. Add greens, mint, 2 tablespoons oil, and 2–3 tablespoons reserved liquid; toss to combine.
Place burrata on plates, top with salad, and drizzle with oil. Season with salt and pepper.