1. Prepare the Chicken Marinade:
* In a medium bowl, combine the olive oil, lemon juice, red wine vinegar, minced garlic, dried oregano, dried thyme, smoked paprika (if using), salt, and black pepper. Whisk well to combine.
* Add the cut chicken pieces to the marinade, ensuring each piece is well-coated.
* Cover the bowl and refrigerate for at least 30 minutes, or ideally for 2-4 hours to allow the flavors to penetrate deeply. Overnight marination is even better for maximum flavor.

2. Make the Tzatziki Sauce:
* Grate the cucumber using the large holes of a box grater. Place the grated cucumber in a clean kitchen towel or several layers of paper towels and squeeze out as much excess liquid as possible. This step is crucial for a thick, creamy tzatziki and prevents it from becoming watery.
* In a separate bowl, combine the squeezed cucumber, Greek yogurt, minced garlic, lemon juice, chopped dill, chopped mint (if using), olive oil, and salt.
* Stir everything together until well combined. Taste and adjust seasoning if necessary (more salt, lemon, or garlic).
* Cover and refrigerate for at least 30 minutes to allow the flavors to meld. Tzatziki tastes even better when made a few hours ahead.

3. Prepare the Vegetables:
* While the chicken is marinating and the tzatziki is chilling, prepare your fresh toppings.
* Dice the tomatoes and cucumber. Thinly slice the red onion (and soak if desired). Halve the Kalamata olives. Crumble the feta cheese. Chop the parsley. Shred the lettuce if using.
* Keep these ingredients separate or combine them into a simple Greek-style salad if preferred.

4. Cook the Chicken:
* You have several options for cooking the chicken:
* Stovetop (Skillet/Grill Pan): Heat 1 tablespoon of olive oil in a large skillet or grill pan over medium-high heat. Once hot, add the marinated chicken pieces in a single layer (you may need to do this in batches to avoid overcrowding the pan, which can steam the chicken instead of browning it). Cook for 4-6 minutes per side, or until golden brown, slightly charred, and cooked through (internal temperature should reach 165°F or 74°C).
* Outdoor Grill: Preheat your grill to medium-high. If using wooden skewers, soak them in water for 30 minutes prior to grilling. Thread the chicken onto skewers. Grill for 8-12 minutes, turning occasionally, until cooked through and nicely charred.
* Oven Broiler: Preheat your broiler. Arrange chicken pieces in a single layer on a baking sheet lined with foil. Broil for 5-7 minutes per side, watching carefully to prevent burning, until cooked through and slightly charred.

5. Warm the Pita Bread:
* Warming the pitas makes them softer and more pliable, preventing them from tearing when stuffed.
* Oven: Wrap pitas in foil and warm in a 350°F (175°C) oven for 5-7 minutes.
* Microwave: Wrap pitas in a slightly damp paper towel and microwave for 20-30 seconds until warm and soft.
* Grill/Pan: Briefly warm pitas on the grill or a dry skillet for about 30 seconds per side.

6. Assemble Your Greek Pita Pockets:
* Carefully cut off the top third of each warm pita bread to create a pocket, or simply fold it in half like a taco.
* Spread a generous spoonful of tzatziki sauce inside each pita pocket.
* Layer in the cooked chicken.
* Add your desired fresh vegetables: diced tomatoes, cucumbers, sliced red onions, and shredded lettuce.
* Sprinkle with Kalamata olives and crumbled feta cheese.
* Garnish with fresh chopped parsley.
* Serve immediately with extra tzatziki sauce on the side and lemon wedges for squeezing.