Prepare the Marinade: Mix lemon juice, red wine vinegar, minced garlic, chili flake, Greek yogurt, olive oil, salt, sweet paprika, onion powder, pepper, dried oregano, ground coriander, and cumin in a bowl.
Marinate the Chicken: Place the pounded chicken thighs in a mixing bowl, pour the marinade over the chicken, and mix well. Cover and refrigerate for a few hours or overnight.
Prepare for Cooking: Stack the marinated chicken thighs on a skewer setup as described, using either a special pan or a DIY setup with a potato or onion and wooden skewers.
Cook the Chicken: Cook in a preheated 400°F oven for 60 to 90 minutes, or until done. Baste with juices halfway through.
Make the Tzatziki Sauce: Combine diced cucumber, lemon juice, olive oil, Greek yogurt, minced garlic, dill, salt, and pepper. Mix well and set aside.
Make the Greek Vinaigrette: Whisk together red wine vinegar, lemon juice, olive oil, dried oregano, and Dijon mustard.
Assemble and Serve: Dress the salad with the Greek vinaigrette. Warm flatbreads in a grill pan with a bit of olive oil. Slice the gyro meat and assemble the wraps with the meat, tzatziki sauce, and dressed salad.