Marinate the chicken: In a large mixing bowl, combine 2 tbsp olive oil, 2 tbsp lemon juice, 2 cloves minced garlic, 1 tsp dried oregano, 1 tsp cumin, salt, and pepper. Whisk until emulsified. Add 1 lb chicken breasts cut into strips and toss to coat evenly. Cover and marinate for at least 30 minutes, allowing the flavors to soak in.
Grill the chicken: Heat a medium grill pan over medium-high heat. Once hot, add the marinated chicken strips and cook for 5-6 minutes on each side, or until the internal temperature reaches 165°F (75°C) and the chicken is golden brown. Remove from heat and let rest.
Prepare the tzatziki: In a bowl, combine 1 cup Greek yogurt, 1/2 cup crumbled feta, 1 small grated cucumber (squeeze out excess moisture), 2 cloves minced garlic, 2 tbsp fresh dill, 1 tbsp lemon juice, salt, and pepper to taste. Stir well and refrigerate until needed to allow flavors to blend.
Assemble the gyros: Warm 4 pita or flatbreads on the grill or stovetop for a few seconds on each side. On each pita, layer with 1 cup chopped lettuce, the grilled chicken, 1/2 cup diced tomatoes, and 1/4 cup diced red onion. Drizzle generously with the prepared tzatziki.
Garnish with fresh parsley before serving and enjoy your delicious Mediterranean Chicken Gyros!