Marinate the chicken. In a bowl, whisk yogurt, olive oil, lemon juice and zest, garlic, oregano, cumin, salt, pepper, and paprika if using.
Add chicken and coat well. Cover and refrigerate for at least 30 minutes and up to 8 hours.
Prep the cucumber for tzatziki. Grate the cucumber on the large holes of a box grater. Sprinkle with a pinch of salt and let sit 5–10 minutes.
Squeeze out excess water with a clean towel or your hands until mostly dry.
Make the tzatziki. In a bowl, combine yogurt, drained cucumber, lemon juice, olive oil, garlic, dill, salt, and pepper. Stir until creamy. Adjust lemon and salt to taste.
Chill until ready to serve.
Warm the pita. Wrap pitas in foil and warm in a 325°F (165°C) oven for 5–10 minutes, or toast on a dry skillet for 30–60 seconds per side.
Cook the chicken. Heat a large skillet or grill pan over medium-high. Add a light drizzle of oil. Cook chicken in a single layer, shaking off excess marinade, 3–5 minutes per side until browned and cooked through.
Work in batches to avoid steaming.
Rest and slice. Transfer cooked chicken to a plate, rest 3 minutes, then slice or chop into bite-size pieces.
Assemble gyros. Spread tzatziki on warm pita. Add chicken, tomatoes, red onion, and cucumber. Finish with herbs, feta, and a squeeze of lemon.
Fold and enjoy.