In a large bowl, gently blend together ground lamb, salt, pepper and garlic; form mixture into patties that are 2 inches in diameter and 1/2 inch thick.
Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Add lamb patties, in batches; cook for 2 to 3 minutes per side or until golden brown on the outsides and cooked to desired level of doneness on the insides.
Meanwhile, prepare aioli: In a blender, process egg and lime juice then slowly drizzle in remaining 1/2 cup of olive oil; blend until emulsified and creamy. Add crushed red pepper flakes, gingerroot, cilantro and additional salt and pepper to taste.
Serve lamb patties in individual lettuce leaves topped with sliced red onion and aioli.