In a medium skillet over medium heat, heat oil. Add onion; sauté about 7 minutes until softened and golden brown. Remove 2/3 of the onion; set aside. Chop remaining onion; return to pan over medium heat. Add garlic; sauté 1 more minute until garlic is fragrant. Remove to a large bowl; cool. 

When onion and garlic mixture is cool, add lamb, olives, paprika, salt and pepper. Use your hand to mix gently but thoroughly; form into 6 even patties. Use your thumb to make an indent in the middle of each burger; as it cooks and puffs in the middle it will end up flat, not rounded. 

For the Romesco-ish Sauce: In a small food processor or blender, combine mayonnaise, red pepper, vinegar, paprika, cayenne, salt and pepper; process until smooth. 

Either preheat grill to medium-high, or very lightly oil a nonstick skillet and heat over medium-high heat. Cook burgers either in the pan or on the grill for about 4 minutes per side, until they have reached the desired doneness. (USDA recommends an internal temperature of 160°F.) Top with Manchego if desired; either close grill lid or cover the pan until cheese starts to melt. 

Toast buns if desired. Place a burger on the bottom of each bun, top with Romesco-ish sauce, and reserved sautéed onions. Serve hot.