Heat the olive oil in a medium skillet over medium heat, and add the onion. Sauté for about 7 minutes until the onion is softened and golden brown. Remove about 2/3 of the sautéed onion and set aside. Chop the rest of the onion, then return to the pan over medium heat. Add the garlic to the pan and sauté for one more minute until you can smell the garlic. Remove to a large bowl and cool.
When the onion and garlic mixture is cool add the lamb, olives, ½ teaspoon smoked paprika and salt and pepper. Use your hand to mix gently but thoroughly. Form the mixture into 6 even patties, and use your thumb to make an indent in the middle of the burger so that as it cooks and puffs in the middle it ends up flat and not rounded.
Make the sauce. Combine the mayo, roasted peppers, paprika, cayenne, sherry vinegar or lemon juice, and salt and pepper in a small food processor or blender. Process until smooth.
Either preheat the grill to medium high, or very lightly oil a nonstick skillet and heat over medium high heat. Cook the burgers wither in the pan or on the grill for about 4 minutes per side, until they have reached the desired doneness. The USDAS recommends an internal temperature of 160°F . Top with Manchego at the end if desired and either close the grill lid or place a lid over the pan until the cheese starts to melt.
Toast the buns if desired, again either on the grill or in a pan on the stovetop. Place a burger on the bottom of each bun, top with the romesco-ish sauce, and then top with the reserved sautéed onions. Serve hot.