Prepare the Marinade and Chicken: In a large resealable plastic bag, combine the minced garlic, white wine vinegar, lemon juice, olive oil, Greek yogurt, dried oregano, salt, and pepper. Seal the bag and gently massage the mixture to blend all the flavors. Add the chicken thighs to the bag, ensuring each piece is thoroughly coated. Seal the bag tightly and refrigerate for at least 2 hours to allow the flavors to infuse—ideally 3 hours, or up to 12 hours for maximum tenderness. Avoid marinating longer than 24 hours to maintain texture.
Make the Tzatziki Sauce: Cut the cucumbers in half lengthwise and use a spoon to scrape out the watery seeds. Grate the cucumbers coarsely using a box grater. Place the grated cucumber into a clean kitchen towel or paper towels and squeeze firmly to remove excess moisture—this step prevents watery tzatziki. Transfer the drained cucumber to a mixing bowl, then add the Greek yogurt, lemon juice, olive oil, minced garlic, salt, and pepper. Stir until well combined. Cover and refrigerate for at least 20 minutes to let the flavors meld beautifully.
Prepare the Fresh Salad: In a medium bowl, combine the diced tomatoes, cucumbers, finely chopped red onion, and fresh parsley if using. Season lightly with salt and pepper, then toss gently to mix. Set aside to complement the gyros with a fresh crunch.
Cook the Chicken: Heat a grill to medium-high heat and lightly brush with oil to prevent sticking. Alternatively, warm a tablespoon of olive oil in a frying pan over medium-high heat. Remove chicken from the marinade, shaking off excess, and place it on the grill or in the pan. Cook each side for about 2 to 3 minutes until the chicken develops a golden-brown sear and is cooked through (internal temperature should reach 165°F/74°C). Remove the chicken from heat and transfer to a plate. Loosely cover with foil and allow it to rest for 5 minutes to retain juices.
Assemble the Gyros: If chicken thighs are large, slice them into manageable strips. Warm pita or flatbreads briefly in a pan or oven. Lay each flatbread on a piece of parchment or foil for easy handling. Spoon a generous amount of the salad onto the center of the bread, layer on the sliced chicken, then finish with a dollop of the chilled tzatziki sauce. Roll the bread tightly around the filling, twisting the parchment paper ends to secure the wrap. Cut in half if preferred, and serve immediately.