In a large bowl, add ground lamb, 1½ cloves of the minced garlic, cumin, coriander, paprika, allspice, cinnamon, salt, black pepper, and olive oil. Mix gently until just combined. Do not overwork.
Divide the lamb mixture into 4 equal portions and shape into patties slightly wider than your buns. Set aside.
In a separate bowl, whisk together Greek yogurt, mayonnaise, lemon juice, lemon zest, remaining garlic, cumin, smoked paprika, cayenne (if using), and salt until smooth. Refrigerate while cooking the burgers.
Heat a skillet or grill pan over medium-high heat. Cook the lamb patties for 4–5 minutes per side, until deeply browned and cooked through (about 160°F internal temperature).
Toast buns or warm pita if desired.
Spread the garlic sauce on the buns or pita, add the lamb patty, then top with tomato, cucumber, pickled onions, and fresh herbs. Serve immediately.