For the tzatziki: Whisk together the yogurt, sour cream, dill, cucumber, garlic, 1/2 teaspoon salt and a few grinds of black pepper in a small bowl until smooth. Set aside.
For the burgers: Mix the beef with the cilantro, mint, coriander, cumin, red pepper flakes, garlic, 1 teaspoon salt and a few grinds of black pepper in a large bowl until well combined. Form the meat into 4 balls, then press into 4-inch patties (each 1 inch thick). Make a 2-inch-wide indentation on the top of each patty with your thumb; this will prevent the burgers from bulging up during cooking.
Set up a breading station: Divide the flour between 2 shallow bowls and lightly beat the egg in a third bowl. Working with one piece of cheese at a time, moisten the cheese under running water, then dip in the first bowl of flour, shaking off the excess. Dip in the egg, then the second bowl of flour. Set aside on a plate.
Heat 1 tablespoon of the oil over medium-high heat in a large skillet. Add the beef patties and cook undisturbed until browned, 2 to 3 minutes. Flip and continue cooking until browned and cooked to desired doneness, 2 to 3 minutes more for medium. Set aside on a plate.
Wipe out the skillet, add the remaining 2 tablespoons oil and heat over medium heat. Add the cheese and cook until golden, about 2 minutes. Flip and cook until golden and heated through, about 3 minutes more. Remove the skillet from the heat. Carefully pour the ouzo over the cheese and immediately light with a long stick lighter. Let the flames burn for 10 to 15 seconds (yell out "OPA!"), then squeeze the lemon halves over the top to put out the flames.
Spread the tzatziki on the cut sides of the buns. Top the bottom half of each bun with a burger, a cheese slice, some tomato, onion, cucumber and olives and the other piece of bun.
Kasseri is a slightly tart firm white cheese. Haloumi is a good substitute for this dish
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