In a large mixing bowl, combine the ground lamb, chopped parsley, chopped mint, grated onion, minced garlic, cumin, coriander, cayenne pepper, salt, and black pepper.

Gently mix the ingredients with your hands until just combined, being careful not to overwork the meat to ensure the burgers stay tender.

Divide the mixture into 4 equal portions and shape them into patties about 3/4-inch thick, making a slight indentation in the center of each to prevent bulging.

Preheat a grill or large cast-iron skillet over medium-high heat and lightly oil the grates or pan surface.

While the grill heats, prepare the tzatziki by mixing the Greek yogurt, squeezed cucumber, and lemon juice in a small bowl; season with a pinch of salt.

Place the patties on the hot grill or skillet and cook for 4 to 5 minutes per side, or until browned and the internal temperature reaches 160°F (71°C).

During the last minute of cooking, place the cut sides of the brioche buns on the grill or pan to toast until golden brown.

To assemble, place a lettuce leaf and tomato slice on each bottom bun, followed by the cooked lamb patty.

Top each patty generously with the prepared tzatziki sauce and cover with the top bun.

Serve immediately while hot, alongside sweet potato fries or a fresh Greek salad for a complete meal.