In a bowl, combine the ground beef with the minced garlic, Worcestershire sauce, smoked paprika, black pepper, and a pinch of salt. Mix thoroughly until well combined. Shape the mixture into 4 large patties.
Heat the olive oil in a skillet over medium heat. Cook the patties for about 4-5 minutes per side, until they are nicely browned and cooked through.
While the meat is cooking, slice the tomatoes, red onion, and prepare the lettuce leaves.
Halve the hamburger buns and toast them in a dry skillet or oven until lightly crispy on the outside but still soft on the inside.
Place a lettuce leaf on the bottom half of each toasted bun, then top with a hot beef patty. Sprinkle with cheddar cheese, allowing it to melt slightly from the heat.
Pile on the tomato slices, red onion, and then generously spoon over the gyro sauce. Top with the other half of the bun and serve immediately!
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