Preheat oven to 300°.
Place rosemary and garlic in a food processor; process until finely chopped. Sprinkle lamb evenly with salt and pepper; rub surface of roast with garlic mixture. Heat oil in a large Dutch oven over medium-high heat. Add lamb to pan, cook 7 minutes, turning to brown on all sides. Stir in broth and vermouth; bring to a simmer. Cover and bake at 300° for 2 1/2 hours.
Add potatoes to pan. Cover and bake an additional 30 minutes or until potatoes are tender. Remove lamb from pan. Cover and keep warm. Strain potato mixture through a sieve over a bowl. Reserve potatoes, and keep warm. Return cooking liquid to pan; bring to a boil over medium-high heat. Cook 10 minutes or until reduced to 3 cups. Add cornstarch to pan, stirring with a whisk; bring to a boil. Cook 1 minute, stirring constantly.
Shred lamb with 2 forks. Serve with potatoes and sauce.