Trim and clean the lamb backstrap. Coat the backstrap with oil, paprika, cumin, coriander, salt, and pepper.
Heat a pan or barbecue grill to high heat. Drizzle a tablespoon of olive oil into the pan. Sear the lamb strips for approximately 2–3 minutes on each side or until they are nicely caramelised on the outside and pink on the inside. Remove from the heat and allow to rest briefly.
In a separate bowl, combine the finely chopped parsley and sumac. Add a drizzle of olive oil, red onion, and a little salt. Massage the mixture with your fingers to soften the onions and allow the flavours to meld.
After resting, cut the lamb into long, thin strips suitable for wrapping.
Lay the flatbreads on a clean surface. Spread a generous amount of black garlic aioli over each.
Place a portion of the cooked lamb in the centre, then top with a spoonful of the onion and parsley salad.
Fold the wraps tightly. Toast in a pan or sandwich press until golden and crisp, allowing the base to absorb any residual meat juices. Serve immediately while warm.