Combine the gyro marinade: In a large mixing bowl, combine the Greek yogurt, olive oil, lemon juice, minced garlic, dried oregano, paprika, and salt. Stir until fully blended.
Prepare the chicken: Trim any excess fat from the chicken thighs. If needed, lightly pound thicker parts of the chicken thighs for even thickness.
Marinate the chicken: Add the chicken thighs to the marinade, turning to coat each piece thoroughly. Cover the bowl tightly with plastic wrap or transfer the chicken and marinade to a large zip-top bag. Refrigerate for at least 4 hours, ideally overnight for the best flavor and tenderness.
Make the Tzatziki Sauce: Peel (optional) and grate the cucumber, then place it into a clean kitchen towel or cheesecloth and squeeze out as much moisture as possible. In a medium bowl, combine the strained cucumber, Greek yogurt, minced garlic, fresh lemon juice (or red wine vinegar), fresh dill, salt, and pepper. Stir well until fully combined. Cover and refrigerate the tzatziki until ready to serve, allowing the flavors to meld. Tip: Homemade tzatziki can be made up to 2 days ahead.
Preheat the oven to 375°F (190°C). Set up the vertical spit on a baking sheet or roasting pan.
Assemble the chicken: Thread the marinated chicken thighs onto the spit one at a time, stacking them tightly and rotating the direction of each thigh a bit to build an even, stable column. Press down occasionally to compact the stack and promote even cooking.
Roast the chicken: Place the vertical spit on the lower rack of your preheated oven. Roast for 1 to 1.5 hours, depending on the thickness of your stack. (For a 2.5pounds of meat, expect about 90 minutes.) After about 45 minutes, tent the top of the meat with a small piece of aluminum foil to prevent over-browning and baste the chicken with leftover marinade or juices collected at the bottom of the pan once or twice during cooking. The chicken is fully cooked and ready when the internal temperature registers 165°F (74°C) at the thickest point as measured by a digital meat thermometer. Rest, then carve the chicken gyros: Once cooked, remove from the oven and allow the chicken to rest for 5 minutes. Using a sharp knife, shave off thin slices of the crispy outer layers.
Assemble the gyro pitas: In the meantime, brush each pita bread lightly with olive oil and warm in a dry skillet or griddle over medium heat. Warm for about 30-45 seconds per side. Top each pita with sliced chicken gyros, a generous spoonful of tzatziki sauce, tomato wedges, cucumber slices, red onion, and feta cheese.