01 - Whisk together olive oil, lemon juice, garlic, oregano, salt, and pepper in a bowl. Add chicken breasts, turning to coat evenly. Marinate for at least 15 minutes, up to 1 hour for maximum flavor infusion.
02 - Combine Greek yogurt, olive oil, lemon juice, grated and drained cucumber, minced garlic, chopped dill, salt, and pepper in a small bowl. Mix thoroughly until smooth and creamy. Refrigerate until ready to serve.
03 - Preheat grill pan or skillet over medium-high heat. Cook marinated chicken breasts for 5 to 6 minutes per side until internal temperature reaches 165°F and juices run clear. Transfer to cutting board, let rest for 5 minutes, then slice thinly against the grain.
04 - Heat dry skillet or oven to medium-low. Warm flatbreads for 1 to 2 minutes until soft and pliable, taking care not to crisp or dry them out.
05 - Spread generous layer of tzatziki sauce over each warmed flatbread. Layer baby spinach or arugula, sliced grilled chicken, cherry tomatoes, cucumber, red onion, Kalamata olives, and crumbled feta cheese.
06 - Drizzle additional tzatziki sauce over toppings if desired. Serve immediately while flatbreads are still warm and chicken is at peak tenderness.