Make the tzatziki first. In a bowl, combine Greek yogurt, grated and squeezed cucumber, garlic, lemon juice, dill, olive oil, salt, and pepper. Stir until creamy.
Taste and adjust seasoning. Refrigerate while you cook the burgers so the flavors meld.
Mix the burger ingredients. In a large bowl, add ground turkey, red onion, garlic, parsley, dill, oregano, lemon zest, lemon juice, feta, breadcrumbs (if using), egg (if using), salt, and pepper. Use your hands or a spatula to gently mix until just combined.
Don’t overwork the meat.
Form the patties. Divide the mixture into 4–6 equal portions and shape into 1/2- to 3/4-inch-thick patties. Press a slight dent in the center of each patty with your thumb to prevent puffing.
Cook on a skillet or grill. Skillet: Heat 1–2 tablespoons olive oil over medium to medium-high heat. Cook patties 4–5 minutes per side, until deeply browned and the internal temperature reaches 165°F/74°C.
Grill: Oil grates and preheat to medium-high.
Grill 4–5 minutes per side, flipping once, to 165°F/74°C.
Oven: Bake on a lined sheet at 400°F (205°C) for 15–18 minutes, then broil 1–2 minutes for color, to 165°F/74°C.
Let them rest. Transfer patties to a plate and rest for 3–5 minutes so juices redistribute.
Assemble. Toast buns if using. Add greens, the turkey patty, a generous spoonful of tzatziki, and sliced cucumbers, tomatoes, and red onion. Serve extra tzatziki on the side.