Crumble ground meat into a large bowl and combine with breadcrumbs, onion, garlic, and all herbs and seasonings until all ingredients are evenly distributed throughout the meat mixture.
Separate the meat into three equal-sized chunks and press firmly to flatten into burger patties. Ensure each patty is slightly wider than your hamburger bun.
Prepare the tzatziki sauce by squeezing the excess liquid out of the grated cucumber and then combine everything in a small bowl with Greek yogurt, white wine vinegar, lemon juice, dill, olive oil, salt, and ground black pepper.
Place patties in a large skillet set over medium heat, working in batches if needed. Cook patties for 2-4 minutes per side, or until the burger is cooked through.
While the burger is cooking lightly toast the interior of your hamburger buns.
Assemble your gyro burgers by adding a leaf of lettuce to the bottom bun and top with gyro patty, onion, tomato, and tzatziki sauce. Serve immediately.