How to Make It
In a large pot, combine the chicken broth and beans with 4 cups of water and bring to a boil. Simmer over moderate heat until the beans are tender, about 1 hour.
Meanwhile, in a large skillet, heat the olive oil. Add the duck legs and cook over moderate heat, turning occasionally, until crisp and heated through, about 15 minutes. Transfer to a plate to cool slightly, then coarsely chop or shred the meat. Set aside the crispy skin.
Add the shallots, carrots, garlic, thyme sprigs and herbes de Provence to the skillet and cook over moderate heat, stirring occasionally, until the shallots are caramelized and completely tender, about 20 minutes. Add 2 tablespoons of water to the skillet and stir, scraping up any browned bits. Discard the thyme sprigs. Stir the shallots and carrots into the beans, then stir in the duck and radicchio. Garnish with thyme leaves and serve topped with the crispy skin.