Step 1: Marinate the Chicken: In a large bowl, combine olive oil, lemon juice, garlic, oregano, paprika, cumin, salt, and pepper. Add chicken and coat well. Cover and refrigerate for at least 1 hour (up to overnight).
Step 2: Make the Tzatziki Sauce: Grate cucumber, squeeze out excess water using a cheesecloth or paper towel. In a bowl, mix yogurt, cucumber, garlic, olive oil, lemon juice, dill, salt, and pepper. Chill in the refrigerator until ready to use.
Step 3: Cook the Chicken: Heat a grill pan, skillet, or outdoor grill over medium-high heat. Cook chicken for 5–6 minutes per side until golden brown and cooked through (internal temperature 165°F / 74°C). Let rest 5 minutes, then slice into strips.
Step 4: Warm the Pita: Heat pita bread on a dry skillet for 1–2 minutes per side, or wrap in foil and warm in oven.
Step 5: Assemble the Gyros: Place warm pita on a plate. Add lettuce, chicken strips, tomato, onion, and feta (if using). Spoon over tzatziki sauce generously. Fold or wrap, then garnish with fresh parsley or dill.