How to Make It
Place the pepper rings in a small heatproof bowl. In a small saucepan, boil the vinegar and salt, stirring, until the salt dissolves; pour over the pepper rings. Let stand at room temperature for 90 minutes. Drain.
In a small bowl, combine the vinegar and shallot and let stand for 30 minutes. Slowly whisk in both oils until incorporated. Season with salt and pepper.
In a food processor, puree 1 ball of burrata until smooth and spreadable. Spread the burrata puree in shallow bowls. In a medium bowl, toss the mint, pea tendrils, purslane and pickled pepper rings with the dressing. Season with salt and pepper and toss to coat. Mound the salad in the bowls. Tear the remaining burrata into small pieces and arrange in the salads along with the peach wedges.