In a large bowl, combine the parsley, yogurt, garlic, coriander, salt, cumin, paprika, and allspice. Add the lamb and mix gently by hand until evenly combined. Shape the meat into 4 patties, each about 4 by 6-inches by 1/2-inch.
Oil 1 side of the pizza dough very lightly. Grill, oiled side down, over medium-high heat until the bottom side is blistered. Lightly oil the top and flip over. Scatter the cheese over the dough and cook until the underside is blistered. Move to indirect heat, cover, and cook until the cheese is melted and the dough is cooked through, about 3 to 5 minutes. Cut the dough into rough rectangles about the size of the patties and keep warm while you grill the burgers.
Lightly oil the patties and arrange them on the hot side of the grill. Cook for 2 to 3 minutes, flip, and cook another 2 to 3 minutes for medium-rare to medium. Transfer the burgers to the bread, top with some Muhamarra and grilled onions and top with another piece of the grilled pizza dough. Put them back on the grill over low or indirect heat for a few seconds per side. Remove them from the grill, halve the sandwiches on a long diagonal, and serve immediately.
Put all of the ingredients, except the lemon zest, in a food processor and process until a smooth paste forms. (Start with 1 hot pepper, taste, and then add more if you want it hotter.) If the mixture is too thick, drizzle in a little more olive oil. Stir in the lemon zest by hand. Taste and add more salt and hot pepper, if desired.