Whisk together the vinegar and sugar until the sugar has dissolved. Stir in the cucumber, onion, chilli and dill and leave to pickle for half an hour, turning halfway.
Combine the yogurt, tahini, lemon zest and juice in a small bowl. Spread over the breads and divide the shredded chicken between them. Top with half the pickle, roll up into wraps and serve with the remaining pickle on the side.