01 - Whisk together olive oil, lemon juice, minced garlic, dried oregano, salt, and black pepper in a mixing bowl. Add chicken breasts, turning to coat evenly. Let marinate for at least 15 minutes, up to 1 hour for deeper flavor penetration.
02 - Combine Greek yogurt, olive oil, lemon juice, grated and drained cucumber, minced garlic, chopped fresh dill, salt, and pepper in a small bowl. Mix thoroughly until smooth and well incorporated. 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 and let rest for 5 minutes before slicing into thin strips.
04 - Heat dry skillet or oven to medium-low. Warm flatbreads for 1 to 2 minutes until soft and pliable, being careful not to overcook or crisp.
05 - Spread generous layer of tzatziki sauce over each warmed flatbread. Layer with baby spinach or arugula, sliced grilled chicken, halved cherry tomatoes, sliced cucumber, red onion slices, 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 optimal temperature.