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 serving.
02 - Mix ground chicken, breadcrumbs, feta, red onion, parsley, garlic, oregano, cumin, salt, and pepper in a large bowl. Combine just until incorporated, avoiding overmixing to maintain texture.
03 - Divide mixture into 4 equal portions and shape into burger patties, pressing gently to form compact discs without compacting too tightly.
04 - Heat olive oil in a large skillet or grill pan over medium heat. Cook patties 5–6 minutes per side until golden brown and internal temperature reaches 165°F.
05 - Place lettuce on bottom half of each toasted bun. Add chicken patty, tomato slices, red onion, and generous tzatziki. Top with remaining bun half and serve immediately.