1. Combine garlic, 5 tablespoons yogurt, and next 4 ingredients (through chickpeas) in a bowl; mash with a potato masher or fork until almost smooth or to desired consistency. Stir in cucumber and dill.
2. Combine remaining 1 tablespoon yogurt and carrot in a small bowl; stir well to coat.
3. Line each pita half with 1 lettuce leaf; fill with 1/2 cup chickpea mixture and 2 tablespoons carrot mixture.