01 - Combine Greek yogurt, grated cucumber, dill, garlic, lemon juice, olive oil, salt, and pepper in a medium bowl. Stir thoroughly, cover, and refrigerate until ready to serve.
02 - Combine ground chicken, breadcrumbs, feta, red onion, parsley, garlic, oregano, cumin, salt, and pepper in a large bowl. Mix just until incorporated, avoiding overmixing to prevent tough patties.
03 - Divide the chicken mixture into 4 equal portions and shape each into uniform patties approximately 3/4-inch thick.
04 - Heat olive oil in a large skillet or grill pan over medium heat. Cook patties for 5-6 minutes per side until golden brown and cooked through, reaching an internal temperature of 165°F.
05 - Place lettuce on the bottom half of each toasted bun. Top with a chicken patty, tomato slices, red onion, and a generous spoonful of tzatziki sauce. Cover with the top bun.
06 - Serve immediately with extra tzatziki sauce on the side for additional flavor.