1. Stir together first 4 ingredients and 2 Tbsp. olive oil; stir in shredded chicken. Add table salt and black pepper to taste; let stand 15 minutes.
2. Meanwhile, stir together radishes, onion, and 1 Tbsp. olive oil. Stir together tomatoes and 1 Tbsp. olive oil. Stir together cucumbers and remaining 1 Tbsp. olive oil. Season each mixture with salt and pepper to taste.
3. Layer chickpea mixture, radish mixture, 3/4 cup Yogurt Dressing, tomato mixture, pita chips, and cucumber mixture in a 4-qt. bowl; top with remaining Yogurt Dressing. Cover and chill 2 to 4 hours. Serve with lemon wedges and additional pita chips.
Total time includes dressing.