Prepare the Marinade: In a large mixing bowl, whisk together the non-fat Greek yogurt, lemon zest and juice, 2 tablespoons of the extra virgin olive oil, red wine vinegar, minced garlic, oregano, smoked paprika, cumin, coriander, black pepper, and optional cayenne.
Marinate the Chicken: Add the chicken strips to the bowl with the marinade, tossing thoroughly to ensure every piece is coated. Cover the bowl and refrigerate for at least 30 minutes or up to 4 hours.
Prep the Sheet Pan: Preheat your oven to 400°F (200°C). In a separate small bowl, toss the rinsed and dried chickpeas with the remaining 1 tablespoon of extra virgin olive oil. Spread the marinated chicken strips in a single layer on one side of a large, rimmed baking sheet. Spread the oiled chickpeas on the other side.
Roast the Chicken and Chickpeas: Place the baking sheet in the preheated oven and roast for 20-25 minutes, flipping the chicken strips halfway through. The chicken is done when it's cooked through and lightly browned (internal temperature of 165°F / 74°C), and the chickpeas are slightly crisped.
Make the Sauce: While the chicken roasts, prepare the sauce. In a small bowl, combine all ingredients for the Dill-Cucumber Yogurt Sauce: yogurt, grated cucumber, fresh dill, minced garlic, and lemon juice. Whisk until smooth, then stir in the olive oil.
Assemble the Gyros: Warm the whole-wheat pitas in the oven for the last 1-2 minutes of cooking or warm them in a dry skillet. To assemble, lay a warm pita flat, spread a generous layer of the yogurt sauce down the middle, top with several strips of roasted chicken, a spoonful of crispy chickpeas, and finish with sliced tomatoes, red bell pepper, red onion, Kalamata olives, and shredded lettuce. Fold and serve immediately.