In a large bowl, combine ground beef, finely grated onion, chopped parsley, allspice, cinnamon, salt, and black pepper. Mix gently with your hands just until evenly incorporated. Do not overmix. This takes about 5 minutes.

Divide the mixture into 4 equal portions and shape each into an oval patty about 4 inches long and 1/2 inch thick. Place on a plate and refrigerate for 10 minutes to firm up.

Add peeled garlic cloves and 1/2 tsp salt to a food processor and pulse until finely minced. With the processor running, drizzle in the oil very slowly in a thin, steady stream. Alternate adding the ice-cold water and lemon juice gradually. Process until the sauce is thick, creamy, and fluffy, about 10 minutes. Adjust consistency with a splash more water if needed.

Heat a non-stick skillet or cast iron pan over medium-high heat and add a small amount of oil. Cook each kafta patty for 4 to 5 minutes per side without pressing down until browned and cooked through to an internal temperature of 160 degrees F (71 degrees C).

While the patties cook, warm the pita breads in a dry skillet or wrapped in foil in a 350 degree F (175 degrees C) oven for 5 minutes until soft and pliable.

Spread a generous layer of toum on each warm pita. Place a hot kafta patty on top and add sliced tomato and cucumber. Add pickled turnips or onions if desired. Garnish with fresh mint or parsley, fold or roll the pita around the filling, and serve immediately.