In a medium bowl, whisk together the olive oil, lemon juice, minced garlic, oregano, salt, pepper, and smoked paprika. Add the sliced chicken to the bowl and toss until every piece is thoroughly coated. Cover the bowl and let the chicken marinate in the refrigerator for at least 30 minutes, or up to 4 hours for best results.

While the chicken is marinating, grate the cucumber and squeeze out excess water using your hands or a clean kitchen towel. In a separate bowl, combine the squeezed cucumber, Greek yogurt, chopped dill, lemon juice, and minced garlic. Stir until well combined. Season with salt and pepper to taste. Cover and refrigerate until ready to serve.

Heat a large skillet or grill over medium-high heat. Add a splash of olive oil if using a skillet. Add the marinated chicken in a single layer; cook in batches if necessary. Cook for 3-4 minutes per side, until golden brown and cooked through (internal temperature 165°F/74°C).

Warm the pitas by wrapping in a damp paper towel and microwaving for 30 seconds, or placing on a dry skillet over medium heat for 20-30 seconds per side. Chop the tomatoes, cucumber, onion, lettuce, and feta if not already done.

Lay a warm pita flat and spread a generous spoonful of tzatziki sauce down the middle. Top with cooked chicken, shredded lettuce, diced tomatoes, cucumber, thinly sliced red onions, and crumbled feta. Fold the pita in half and serve immediately.