Make the sauce. In a small bowl, stir together Greek yogurt, lemon juice and zest, dill or mint, grated garlic, salt, and pepper. Taste and adjust seasoning.
Refrigerate while you prepare the burgers.
Grate the onion. Use the large holes of a box grater to grate the onion. Don’t drain the juices; they help keep the burgers moist.
Mix the patties. In a large bowl, combine ground turkey, feta, grated onion, garlic, parsley, dill, oregano, lemon zest and juice, olive oil, salt, pepper, and red pepper flakes if using. Gently mix until just combined. Do not overwork or the burgers can turn tough.
Form the burgers. Divide the mixture into 4 large or 5 medium patties, about 1/2 inch thick.
Press a small indent in the center of each patty to prevent doming as they cook.
Chill briefly (optional but helpful). Place patties on a plate, cover, and refrigerate for 15–20 minutes. This helps them hold together better on the grill or in the pan.
Cook on the stovetop. Heat a large nonstick skillet or cast-iron pan over medium heat. Lightly oil the pan.
Cook patties 4–5 minutes per side, until the internal temperature reaches 165°F (74°C).
Or grill them. Preheat the grill to medium-high and oil the grates. Grill 4–5 minutes per side, flipping once, until cooked through.
Rest and assemble. Let the burgers rest for 3 minutes. Serve on buns or in warm pita with yogurt sauce, cucumber, tomato, red onion, and greens.