Start by grating the cucumber and squeezing out as much liquid as possible. This step is important to keep the sauce thick and creamy. Mix the cucumber with Greek yogurt, garlic, lemon juice, and dill. Season with salt and pepper to taste. Place the sauce in the fridge to chill while you prepare the burgers. This helps the flavors blend beautifully.
In a large bowl, combine ground chicken, breadcrumbs, egg, garlic, onion, parsley, oregano, salt, pepper, lemon juice, and olive oil. Mix everything gently until just combined, being careful not to overwork the mixture. This keeps the burgers tender. The mixture should be slightly sticky but easy to shape into patties.
Divide the mixture into equal portions and shape them into round burger patties. Press them gently so they cook evenly without falling apart. Let them rest for a few minutes before cooking to help them hold their shape better. Heat a skillet or grill pan over medium heat and lightly grease it. Cook the patties for about 5–6 minutes on each side until fully cooked and golden brown. Make sure the internal temperature reaches a safe level and the burgers remain juicy inside.
Toast the buns lightly for added texture and flavor. Place lettuce, the cooked chicken patty, tomato, and onion on the bun. Top generously with tzatziki sauce and cover with the top bun before serving.