Braised lamb shanks in a robust tomato sauce and mashed potatoes and parsnips make a hearty meal that is grown-up comfort food, perfect for fall or winter.
Preheat oven to 375 degrees. With a sharp knife, cut an incision 1 inch from narrow end of each shank, cutting to the bone and all the way around (this will help the meat separate from the bone as it cooks).
Heat the oil and butter in an 8-quart Dutch oven or skillet over medium heat. Season shanks well with salt and pepper. Place flour on a plate; dredge shanks one at a time in flour, turning to coat both sides, and shake off excess. Place on a baking sheet. Working in two batches, sear shanks until well browned all over, about 5 minutes per side. Transfer to a large platter; drain off all but 2 tablespoons fat after the last batch.
Return skillet to medium heat; saute celery, onions, and leek until soft and golden, about 6 minutes. Add garlic; cook until soft, about 4 minutes. Add wine; deglaze pan by scraping up any browned bits with a wooden spoon until almost all liquid has evaporated. Return shanks to skillet; add stock, tomatoes, tomato paste, thyme, and bay leaves. Bring to a boil; cover, and transfer to oven to braise 2 hours.
Remove from oven. Using a slotted spoon, transfer lamb and vegetables to a platter. Skim fat from surface of cooking liquid; return skillet to medium heat until liquid is thickened into a sauce, 5 to 7 minutes. Return shanks and vegetables to skillet until heated through. Season with salt and pepper; serve hot.
The dish can be prepared up to 1 day in advance and refrigerated; before serving, reheat lamb and sauce on the stove over medium-low heat for 20 minutes.