01 - Combine Greek yogurt, grated cucumber, dill, garlic, lemon juice, olive oil, salt, and pepper in a medium bowl. Stir thoroughly until well incorporated. Cover and refrigerate until ready to serve.
02 - Place ground chicken, breadcrumbs, feta, red onion, parsley, garlic, oregano, cumin, salt, and pepper in a large bowl. Mix gently until just combined. Avoid overmixing to prevent tough patties.
03 - Divide the chicken mixture into 4 equal portions. Shape each portion into uniform burger patties, pressing lightly to compact without overhandling.
04 - Heat olive oil in a large skillet or grill pan over medium heat. Cook patties for 5 to 6 minutes per side until golden brown and fully cooked through, reaching an internal temperature of 165°F.
05 - Place lettuce leaves on the bottom half of each toasted bun. Top with a cooked chicken patty, tomato slices, red onion rings, and a generous spoonful of tzatziki sauce. Complete with the top bun.
06 - Serve immediately with additional tzatziki sauce on the side for extra flavor.