Mince the garlic in a food processor. Add the chickpeas, shallots, cilantro, cumin, and 1/2 teaspoon of salt and blend until a coarse puree forms. Scrape down the sides of the bowl. With the machine running, gradually add the lemon juice and sesame oil through the feed tube, scraping the sides down as needed. Gradually add the olive oil, blending until smooth and creamy. Slowly drizzle in a little bit of water, 1 teaspoon at a time, until the desired consistency is achieved. Season to taste with more salt.
Mix the cucumber, yogurt, lemon juice, and dried mint in a small bowl. Season to taste with salt.
Toss the onions with the sumac in another small bowl to coat. Season to taste with salt.
Spread the humus over the lavash breads.
Scatter the chicken over the left-hand side of the lavash.
Scatter the lettuce over the chicken. Scatter the radishes and feta cheese over the lettuce.
Scatter the onion mixture over the fillings.
Spoon the yogurt alongside the fillings.
Starting with the short edge alongside the fillings, roll up the lavash. Cut each wrap in half on an angle and serve.