Preheat oven to 350°F (175°C).
Wash and pat dry lamb shanks with paper towel.
Heat 1 tablespoon of oil in a heavy based pot over medium-high heat. Sear two shanks in the hot oil until browned on all sides. Repeat with remaining shanks and oil. Transfer to a plate, tent with foil to keep warm, set aside.
To the pan juices, sauté the carrots and diced onions until softened, (about 3 minutes), then add garlic and cook for 1 minute.
Return the shanks back into the pot; season with 1 teaspoon coarse salt and 1/2 teaspoon ground pepper. Sprinkle with flour, toss well and cook for 4-5 minutes to brown the flour.
Add stock, wine, puree, tomato paste, bouillon and herbs. Bring to a simmer on the stove.
Cover, transfer to lower part of the oven and cook for 2 1/2 hours, or until the meat is fall apart tender (adjust the heat so that the liquid simmers very slowly).
Gently transfer the shanks onto a plate; tent to keep warm.
Discard the bay leaves from the sauce and place pot onto stove. Simmer sauce over medium heat until thickened to your desired consistency. OPTIONAL: skim off any additional fat which rises to the surface. You should be left with about 3 cups of sauce thick enough to lightly coat the back of a spoon.
*If the sauce is too thick, add a few tablespoons of stock. If the sauce is too thin, boil it over medium heat for about 10 minutes, or until reduced to the right consistency. (See notes about thickening.)
Taste for seasoning and adjust salt and pepper, if desired. Add the shanks back into the pot of sauce.
Garnish with parsley and serve with mashed potatoes, rice or pasta.
To serve the following day, allow the casserole to cool completely, cover and refrigerate.
The day of serving, remove from refrigerator for at least an hour before reheating. Place over medium-low heat and let simmer gently for about 10 - 15 minutes, basting the shanks and vegetables with the sauce.