In the bowl of a stand mixer, combine the water, oil, yeast, sugar, and flour. Mix on medium speed for about 3 minutes. Add the salt and continue kneading for 7 minutes more, until the dough is sticky and smooth. Transfer to a greased bowl, cover, and let rise until doubled in size, about 1 hour.

Turn the dough out onto a lightly greased surface and divide into 5 equal pieces. Cover loosely and let rest at room temperature for 15 minutes. Heat a 12-inch/30 cm dry skillet over medium heat. On a floured surface, roll each piece into an 11-inch/28 cm round. Cook each flatbread for about 1½ minutes per side, until puffed and golden with dark spots. Stack the cooked laffas and wrap in a clean towel to keep warm.

Preheat the oven to 250°F/120°C. Place the chicken in a roasting pan and roast for 40 minutes. Remove and let cool to room temperature. Using a sharp knife, slice the meat as thinly as possible.

Heat a large skillet over medium heat and add the lamb fat or oil. Sauté the sliced onion until golden brown, about 8 minutes. Add the chicken, baharat, turmeric, cumin, and salt. Cook for another 5 to 7 minutes, stirring occasionally, until deeply browned and fragrant.

Serve the shawarma tucked into warm laffas with hummus, pickles, and any other toppings you like.