In a large bowl, whisk yogurt, lemon zest/juice, 2 tbsp olive oil, red wine vinegar, garlic, oregano, smoked paprika, cumin, coriander, black pepper, cayenne, and 1 tsp kosher salt.
Add chicken; toss to coat. Cover and refrigerate 30 minutes to 4 hours.
Heat oven to 400°F (200°C). Toss chickpeas with remaining 1 tbsp olive oil and ½ tsp kosher salt.
Shake or wipe excess marinade from chicken; discard leftover marinade. Arrange chicken in a single layer on a rimmed sheet pan. Spread chickpeas on a second sheet pan.
Roast 20–25 minutes, flipping chicken and rotating pans halfway; cook until chicken reaches 165°F (74°C) and chickpeas are lightly crisp.
For the sauce, whisk yogurt, cucumber, dill, garlic, and lemon juice until smooth; stir in olive oil.
Warm pitas in the oven for 1–2 minutes. Spread sauce on each pita, add chicken and chickpeas, top with tomatoes, bell pepper, onion, olives, and lettuce. Fold and serve.