Heat 2 tablespoons olive oil in heavy large Dutch oven over medium-high heat. Add sliced onions and shallots and sauté until brown, about 20 minutes. Mix in 2 tablespoons chopped rosemary. Remove from heat.
Sprinkle lamb shanks with salt and pepper; coat lamb with flour. Heat remaining 2 tablespoons olive oil in heavy large skillet over high heat.
Working in batches, add lamb shanks to skillet and cook until brown on both sides, about 10 minutes per batch. Using tongs, transfer lamb shanks to plate.
Add 1 cup dry red wine to same skillet and bring to boil, scraping up any browned bits.
Pour into Dutch oven with onion mixture. Add remaining 1 1/2 cups red wine, canned beef broth, tomato paste and 2 bay leaves to dutch oven. Bring to boil, stirring until tomato paste dissolves. Add lamb shanks, turning to coat with liquid.
Bring mixture to boil. Reduce heat, cover and simmer until lamb is almost tender, turning lamb shanks occasionally, about 1 1/2 hours. (Can be prepared 1 day ahead. Cover and refrigerate.).
Uncover Dutch oven and boil until liquid is reduced to sauce consistency, stirring and turning lamb shanks occasionally, about 30 minutes. Season with salt and pepper.
Peel potatoes and cut into 2-inch pieces. Peel and halve rutabegas and thinly slice. Peel parsnips and cut into 1-inch pieces.
Bring large pot of salted water to boil. Add potatoes, rutabagas and parsnips. Boil until vegetables are tender, about 30 minutes. Drain well.
Return vegetables to same pot. Mash until coarse puree forms. Mix in 3 tablespoons olive oil. Season vegetables to taste with salt and pepper. (Vegetable mash can be prepared 2 hours ahead. Let stand at room temperature. Rewarm over low heat, stirring frequently.) Transfer vegetables to bowl and serve.
Spoon Potato and Root Vegetable Mash onto plates. Top vegetables with lamb shanks and sauce. Sprinkle lamb shanks with additional chopped fresh rosemary and serve.