Braise the lamb
Position a rack in the center of the oven and heat the oven to 350°F.
Pat the lamb chops dry. Heat the oil in a heavy-duty 12-inch skillet over medium-high heat. Working in 2 batches, cook the lamb chops, flipping once, until well browned on both sides, about 4 minutes total per batch (add more oil for the second batch, if necessary). Season with salt and pepper and transfer to a 9x13-inch roasting pan.
Add 1 cup of the wine to the skillet and bring to a simmer, scraping the skillet with a wooden spatula to loosen any browned bits. Pour the wine over the chops. Sprinkle with the cumin, then add the garlic, carrot, onion, and the remaining 1/2 cup of wine. The liquid should come halfway up the chops; if necessary, add water. Cover the pan with a double thickness of aluminum foil and braise in the oven until the meat is fork-tender, 1-1/2 to 2 hours.
Make the tahini sauce
Meanwhile, combine the garlic and lemon juice in a medium bowl; let sit for 5minutes. Whisk in the yogurt, tahini, olive oil, and 3/4 tsp. salt until smooth. Add 1 to 2 Tbs. water if necessary to achieve a thick yet pourable consistency.
Make the pickled cabbage
Heat the oil in a 10-inch skillet over medium heat. Add the cabbage and cook, stirring occasionally, until just tender, 8 to 10 minutes. Remove from the heat and stir in the pomegranate molasses, vinegar, and sugar. Season to taste with salt, pepper, more vinegar, or sugar.
Finish the lamb
Transfer the lamb chops to a cutting board and tent loosely with foil to keep warm. Strain the contents of the roasting pan through a medium-mesh sieve into a medium bowl (you should have about 2 cups liquid); discard the solids. Freeze the liquid until the fat rises to the surface, about 15 minutes, then skim it off and discard it. (Alternatively, use a fat separator.) Transfer the liquid to a 3-quart saucepan. Boil over medium-high heat until reduced by half, about 10 minutes.
Meanwhile, use your fingers to break the lamb into small chunks, discarding the fat and bones.
Whisk the pomegranate molasses, lemon juice, and butter into the braising liquid. Add the lamb and gently toss to coat. Season to taste with salt and pepper.
Assemble the shawarma
If you're using store-bought flatbreads (or if you made them ahead), warm the flatbreads in a 12-inch skillet (preferably cast iron) over medium heat. Arrange on a clean work surface. Divide the lamb among the flatbreads, spreading it in a strip about 1 inch from the near edge. Top the lamb with a few tablespoons of the cabbage and sauce. Roll the shawarma up tightly, pulling back on the edge of the bread after folding it over the filling and tucking the filling under with your fingertips, to create a slender, burrito-like wrap. Rest the shawarma on their seam side to keep them closed.
Heat the 12-inch skillet over medium heat. Working in batches, cook the shawarma seam side down until browned and crisp on that side (do not flip), about 3 minutes. Serve.
Make Ahead Tips
The sauce may be made up to 2 days ahead; refrigerate and return to room temperature before using.
The cabbage may be made up to 2 days ahead; refrigerate and return to room temperature before using.
The lamb may be made up to 2 days ahead; refrigerate and gently reheat before using.
Serve with pickles and mezze, such as tabbouleh, fattoush, eggplant dip, or a chickpea salad.
nutrition information (per serving):
Photo: Andrew Purcell