Prep the lamb shanks. Pat the lamb shanks dry with a paper towel and season them liberally all over with kosher salt and freshly cracked black pepper. Let them sit at room temperature for 1 hour before cooking. Use a piece of kitchen twine to tie the thyme and tarragon into a bundle.
Preheat an oven to 350°F.
Sear the lamb shanks. Heat the oil in a large braiser or oval dutch oven over medium-high heat. Once the oil is hot and shimmering, sear the lamb shanks all over until very browned. Add the garlic and herb bundle and cook for 1 minute, until the garlic is fragrant. Add the onions, leeks, and carrots and cook for another 2-3 minutes, until the vegetables are starting to soften.
Braise the lamb. Pour in the white wine and bring to a gentle simmer. Season everything with salt and pepper and nestle the lemon slices into the pan. Cover and transfer to the oven for 2 hours. After two hours, add the potatoes and continue to cook for another hour or hour and a half, until the lamb is very tender. Remove from the oven, discard the lemons and herbs, and let the lamb rest in the juices for 20 minutes. Use a large spoon to skim off any excess fat.
Make the gremolata. While the lamb is in the oven, make the gremolata. In a small bowl, mix the herbs, oil, garlic, shallot, and preserved lemon and stir well. Season with salt and pepper and set aside. The gremolata gets better as it sits, so it's best to make it a couple hours ahead of time.