Thinly slice the tomato, red onion, and cucumber. Wash and dry the iceberg lettuce.
In a bowl, combine the ground lamb with minced garlic, oregano, salt, and pepper. Form into two equal-sized patties.
Heat a skillet or grill over medium-high heat. Add a drizzle of olive oil and cook the lamb patties for about 3-4 minutes per side, until nicely browned and cooked through.
Halve the hamburger buns and toast the cut sides in a dry skillet or on the grill for about 1 minute, until lightly golden.
Spread the bottom buns with tzatziki sauce, then top with iceberg lettuce and the cooked lamb patties.
Crumble feta cheese over the patties, then add the tomato slices, red onion, and cucumber.
Sprinkle with fresh dill, then top with the other half of the bun.
Serve immediately while the meat is hot and the flavors are at their best.
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