In a 8-quart Dutch oven, heat vegetable oil over medium-high heat, Sprinkle 1/4 teaspoon salt on lamb shanks. Cook lamb shanks, 2 at a time, until browned on all sides. Remove shanks to a bowl as they brown.
Preheat oven to 350 degrees Fahrenheit. In drippings in Dutch oven, cook onions and carrots until tender and lightly browned, about 10 minutes. Add garlic, cook 2 minutes.
Return lamb shanks to Dutch oven. Add tomatoes with their juice, chicken broth, and 3/4 teaspoon salt; heat to boiling over high heat. Cover Dutch oven and bake 1 1/2 hours. Turn lamb shanks over; add potatoes and green beans. Cover and bake 1 1/4 hours longer or until lamb and potatoes are tender.
Meanwhile, from lemons, grate 1 tablespoon peel and squeeze 2 tablespoons juice.
When lamb shanks are done, skim fat from sauce. Stir in lemon peel, lemon juice, dill, and parsley.