Step 1Heat the oven to 350 degrees. Warm 2 tablespoons of olive oil in a heavy soup pot over medium-low heat. Add the onion and carrots, and cover and cook until they soften, about 20 minutes.
Step 2Trim the branches and fronds from both bulbs of fennel; chop at least one-third cup of the fronds, wrap tightly and refrigerate. Quarter one bulb lengthwise and cut out the solid core. Dice and add to the soup pot. Set the other bulb aside until later.
Step 3When the vegetables in the soup pot are softened and aromatic, stir in the garlic and cook until fragrant, 2 or 3 minutes. Add the beans, bay leaf and 8 cups of water. Cover and place in the oven to cook for 1 hour.
Step 4After 1 hour, remove the pot from the oven and stir in 1 1/2 teaspoons of salt. Return to the oven to finish cooking until the beans are quite tender, another 45 minutes to 1 hour, 15 minutes. Cooking time can vary quite a bit depending on the condition of the beans, so begin checking after 30 minutes.
Step 5When the beans are tender, remove the pot from the oven. If there are still just a few beans that are slightly chalky, that's OK -- leave the pot covered for a while and they will finish cooking from the reserved heat. If the soup loses too much moisture in the oven, add water as needed to maintain a loose, soup-like consistency.
Step 6In a small skillet, heat one-fourth cup of olive oil over medium heat. Quarter the remaining fennel bulb lengthwise, but do not trim the core, so the fennel bulb will stay together. Fry the bulb until well browned on all three sides, covering tightly in between turns to avoid splattering. Remove the pan from the heat momentarily to carefully add the wine, replace the cover, and cook until the fennel is tender, about 10 minutes.
Step 7When the fennel is tender, remove it from the pan, sprinkle with salt and cut each quarter in half lengthwise. Add these to the soup. (The dish can be prepared up to this point a day in advance and refrigerated, tightly covered.)
Step 8When ready to serve, warm the soup over medium heat in a covered pot. Just before serving, stir in the reserved chopped fennel fronds. Add freshly ground black pepper to taste and more salt if necessary. Ladle the soup into warm, wide soup plates and finish each with a drizzle of the best-quality olive oil. Serve immediately.