The night before you want to serve the lamb shanks, liberally salt your meat all over and place the shanks into an air-tight container big enough to fit them with the lid on. Close the lid and refrigerate overnight.
About 5 hours before you want to serve the shanks, remove them from the brining liquid and let them come to room temperature. Once they've warmed up a bit, preheat your oven to 325°F. Dry the meat thoroughly, then sprinkle it with pepper and a little more salt. Heat a large skillet on high heat and add avocado oil. Once the pan is smoking hot, add the lamb shanks. Sear until brown on all sides. Remove from the heat and set aside.
Heat a large dutch oven or braiser on the stove at medium-high heat and add a Tbsp of avocado oil. Once the oil is rippling, add the garlic and onions and cook until tender and fragrant, about 3 minutes. Remove from the heat. Scatter the pieces of preserved lemon all over the onions and garlic, then place the shanks on the top.
Sprinkle the potato starch over the shanks and rub it in a little bit. Drizzle the silan over the tops of the shanks, then sprinkle the za'atar (or herb mixture if you're avoiding kitniyot) on top. Pour the pomegranate juice around the sides of the meat until it comes about half-way up. Braise in the oven, covered. After 1½ hours, remove the lid and turn the shanks over. Return to the oven.
Cook uncovered for another 2 hours, turning the shanks every 30-45 minutes. Once they're finished cooking, remove them from the braising liquid and place onto a serving plate. Garnish with the parsley and pomegranate arils.