01 - Combine Greek yogurt, grated cucumber, minced garlic, dill, lemon juice, olive oil, salt, and pepper in a bowl. Mix thoroughly, cover, and refrigerate until ready to serve.
02 - Combine ground chicken, red onion, garlic, feta, parsley, dill, oregano, cumin, salt, and pepper in a large bowl. Mix gently until just incorporated, being careful not to overwork the meat.
03 - Divide the chicken mixture into 4 equal portions and shape each into uniform patties approximately 1 inch thick.
04 - Heat olive oil in a grill pan or skillet over medium heat. Cook patties for 5-6 minutes per side until golden brown and completely cooked through, reaching an internal temperature of 165°F.
05 - Lightly toast the burger buns or pita breads until golden and slightly crisp.
06 - Layer lettuce on the bun base, add a cooked chicken patty, spoon generous tzatziki over the top, and finish with tomato and onion slices. Crown with the top bun and serve immediately with extra tzatziki on the side.