Make the tzatziki. Grate half a cucumber on the large holes of a box grater. Squeeze out excess water with your hands or a clean towel.
Stir with 1 cup Greek yogurt, 1 grated garlic clove, 1–2 tablespoons lemon juice, 1–2 tablespoons chopped dill, a drizzle of olive oil, and a pinch of salt and pepper. Chill while you prep the burgers.
Prep aromatics. Grate half a small red onion. Finely chop 2 tablespoons parsley and 2 tablespoons dill (or mint).
Zest 1 lemon.
Mix the patties. In a large bowl, combine 1.5 pounds ground turkey, the grated onion, 1 egg, 1/2 cup breadcrumbs, 1/2 cup crumbled feta, 1 teaspoon dried oregano, lemon zest, 1 minced garlic clove, 1 teaspoon salt, and 1/2 teaspoon black pepper. Add 1 tablespoon olive oil for moisture. Gently mix until just combined—don’t overwork.
Shape and chill. Divide into 4–6 patties about 1/2–3/4 inch thick.
Press a shallow dimple in the center of each to prevent puffing. Chill the patties for 15–20 minutes to firm up.
Cook your way. Grill: Preheat to medium-high, oil the grates, and cook 5–6 minutes per side, flipping once, until the internal temperature reaches 165°F.
Stovetop: Heat a film of olive oil in a skillet over medium. Cook 5–7 minutes per side, covering for the last few minutes to help them cook through.
Oven: Bake at 400°F on a parchment-lined sheet for 15–18 minutes, then broil 1–2 minutes to brown.
Check for 165°F.
Toast and assemble. Lightly toast buns or warm pita. Add lettuce, tomato, and sliced red onion. Top each patty with a generous spoonful of tzatziki and, if you like, extra feta and herbs.
Serve. Add cucumber slices or a simple Greek salad on the side.
A squeeze of lemon over the burger before serving brightens everything.