Make the tzatziki: In a medium bowl, combine the Greek yogurt, grated and very well-squeezed cucumber, grated garlic, dill, olive oil, and half of the lemon juice. Stir well to combine. Taste and add more lemon juice as needed. Season generously with salt and pepper. Cover and refrigerate until ready to use.
Make the chicken filling: In a large bowl, combine the shredded rotisserie chicken, ½ cup of the tzatziki, sliced red onion, dried oregano, lemon zest, and olive oil. Toss until the chicken is evenly coated. Taste and season with salt and pepper as needed.
Warm the pitas: Place the pita breads on a baking sheet and warm in a 350 degree F oven for about 5 minutes, until soft, warm, and pliable. Alternatively, warm them directly over a low gas flame for 20 to 30 seconds per side.
Assemble: Carefully open each warm pita to create a pocket and fill generously with the chicken mixture. Top with chopped tomatoes, crumbled feta, chopped Castelvetrano olives, pickled red onions, and fresh dill. Drizzle with additional tzatziki and serve immediately.