Make the burger mixture. Add the ground turkey, chopped spinach, feta, mozzarella, parsley, onion, garlic, panko breadcrumbs, egg, oregano, salt, and black pepper to a large mixing bowl.
Mix gently. Use clean hands or a spatula to gently combine the ingredients until just incorporated. Avoid overmixing, which can make the burgers tough.
Shape the patties. Divide the mixture into 4–6 equal portions and gently shape each portion into a burger patty.
Cook the burgers. Heat a grill pan, skillet, or outdoor grill over medium-high heat. Lightly grease the cooking surface if needed. Cook the turkey burgers for approximately 4–5 minutes per side, or until browned and cooked through.
Check for doneness. Turkey burgers are fully cooked when the center reaches an internal temperature of 165°F.
Serve. Serve immediately on toasted buns with tzatziki, cucumber, tomato, onions, and extra feta, or enjoy the patties on their own with your favorite sides.