In a large bowl, whisk 3 tbsp olive oil, 3 tbsp lemon juice, 3 minced garlic cloves, oregano, cumin, paprika, 1 tsp kosher salt, and 1/2 tsp pepper until combined.

Add chicken and toss to coat thoroughly. Cover and marinate in the refrigerator for at least 1 hour, or up to overnight for best flavor.

Make the feta tzatziki: Grate cucumber, then squeeze out excess liquid using a towel or strainer. In a bowl, stir together Greek yogurt, drained cucumber, 1 minced garlic clove, crumbled feta, 1 tbsp olive oil, 1 tbsp lemon juice, dill, pepper, and a pinch of salt. Chill until serving.

Cook the chicken: Preheat a grill pan, outdoor grill, or skillet over medium-high heat. Remove chicken from marinade and cook 5–6 minutes per side, or until lightly charred and the internal temperature reaches 165°F (74°C).

Transfer chicken to a plate and rest 5 minutes; slice into thin strips.

Warm pitas in a dry skillet, on the grill, or wrapped in foil in a 300°F oven until soft and pliable.

Assemble gyros: Fill each warm pita with lettuce, sliced chicken, tomato, onion, and cucumber. Spoon over a generous amount of feta tzatziki and garnish with parsley or dill. Serve with lemon wedges if desired.