Step 1 Heat the oven to 350 degrees. Cook the garlic sausage in olive oil in the bottom of a Dutch oven over medium heat on the stove top until the sausage has browned, about 7 minutes. Add the carrot and cook until it begins to soften, about 5 minutes. Add the onion and cook, stirring constantly to scrape up any browned sausage bits sticking to the bottom of the pan, until it begins to soften, about 2 to 3 minutes. Add the garlic and cook until fragrant, about 3 minutes.
Step 2 Add the dried beans, water and bay leaf. Bring to a simmer, stirring occasionally. Cover tightly and bake in the oven for 1 hour. Add the salt, stir, and continue cooking, covered, until the beans are tender, 45 minutes to 1 hour. The beans should be fairly dry, but if necessary, add more water, one-quarter cup at a time. Remove the bay leaf and discard. Season to taste with more salt if necessary, and freshly ground black pepper. (The dish can be prepared up to this point a day ahead and refrigerated, tightly covered.)
Step 3When almost ready to serve, chop the dandelion greens in 1-inch sections down to where you're getting almost all stem. You should have 3 to 4 cups.
Step 4While the duck breasts are cooking (below), reheat the beans, adding a little more water if necessary to create a slightly flowing, risotto-like texture, and add the dandelion greens. Cook, covered, until the greens soften, about 5 minutes. Season to taste with more salt and pepper, if necessary, and the red wine vinegar to taste.