In a large bowl, mix the Greek yogurt, olive oil, minced garlic, lemon juice, red wine vinegar, dried oregano, ground cumin, smoked paprika, salt, and black pepper. Whisk thoroughly to combine into a smooth marinade.
Add the thinly sliced pork or chicken to the marinade, ensuring each piece is well coated. Cover the bowl and refrigerate for at least 1 hour, preferably overnight, to allow the flavours to penetrate the meat deeply.
While the meat marinates, prepare the tzatziki sauce. In a medium bowl, combine the Greek yogurt, grated cucumber, minced garlic, lemon juice, chopped dill, and a pinch of salt. Mix well, cover, and refrigerate until ready to serve.
Preheat your oven to 200°C (390°F) or set up a grill to medium-high heat. If using a rotisserie or vertical spit, layer the marinated meat on the spit, pressing down to create a compact stack. Alternatively, thread the meat onto skewers or lay the slices on a baking sheet.
Place the meat in the oven or on the grill, rotating or turning occasionally to ensure even cooking. Roast or grill for 20-30 minutes, or until the meat is golden brown and slightly crispy on the edges.
Once cooked, remove the meat from the heat. Let it rest for a few minutes, then slice it thinly against the grain for the most tender texture. Adjust seasoning if needed.
Warm the pita bread on a dry skillet or directly on the grill for 1-2 minutes on each side, just until soft and pliable. This will help hold the filling better and enhance the overall flavour.
To serve, spread a generous layer of tzatziki on each pita. Add slices of grilled meat, a few tomato slices, onion rings, and lettuce if desired. Garnish with fresh parsley, fold the pita, and serve immediately with extra tzatziki on the side.