Wash and pat dry the lettuce leaves. Wash the tomatoes, pat dry, and cut into thin slices. Peel both onions; finely dice one onion and cut the other into rings.
In a bowl, combine the minced beef, minced lamb, diced onion, grated kefalotiri cheese, gyros spices, and egg. Mix until uniform.
Divide the meat mixture into 6 equal portions and shape each into a patty.
Heat oil in a large pan over medium-high heat. Fry the patties for 3-4 minutes on each side until cooked through (internal temperature 71°C / 160°F).
While the patties cook, cut the hamburger buns in half. Lightly spread tzatziki on the cut side of each bottom half.
Place 2 lettuce leaves on each bottom bun half. Top with a cooked patty, then the Greek cabbage salad, tomato slices, and onion rings. Add a spoonful of tzatziki.
Top with the bun halves and serve with french fries, mayo, and/or ketchup on the side.