For the Braised Lamb Neck:
Preheat the oven to 350˚F. In a large stockpot, heat the oil over medium-high heat. Sear the Lamb neck in batches and set aside. In the same stockpot, sauté the mirepoix until vegetables are caramelized. Off heat, add the red wine, return to heat and reduce to au sec. Add beef stock, fresh herbs, and garlic and bring to a boil. Cover with parchment paper and foil and braise in the oven for 4 hours. Remove from oven, take lamb out of braising liquid, and cool. When cooled, pick meat into large pieces. Reduce the braising liquid, add fresh parsley and sherry vinegar and reserve. Season the lamb with salt and pepper and press ½-inch thick into plastic lined hotel pan. Refrigerate overnight.
For the Beet Purée:
Preheat the oven to 400˚F. Wrap 3 medium-size red beets in foil with garlic, thyme and oil. Roast for 45 minutes or until beets are tender. When slightly cooled, peel off the skin and blend in the bowl of a food processor. Add salt to taste. Pass through chinoise and set aside.
For the Pommes Dauphinoise:
Preheat the oven to 350˚F. Grease and line a half hotel pan with parchment paper. Peel the potatoes and slice paper-thin on a mandolin, lengthwise. In a small bowl, combine the butter, cream, and season with salt and pepper. Spread a small amount of the cream and butter mixture into bottom of hotel pan and cover with a single layer of sliced potatoes. Repeat process until all the potatoes are layered, it should be about ½-inch thick. Bake for 30 minutes or until potatoes are tender.
For the Lamb Loin:
Season lamb loins with salt and pepper. Heat oil in a cast iron skillet over medium-high heat. Sear lamb loins and cook to desired temperature. Rest for 10 minutes before slicing.
To Assemble and Serve:
Warm the braised lamb neck in the oven to heat through. Unmold the potatoes and lamb neck and portion into 3-inch pieces. Spread a large spoonful of beet purée on the bottom of a plate. Place the lamb neck on top of the beet puree. Top with the potato, slices of the lamb loin, and the reserved braised lamb jus.