Pick over the beans to remove any grit. Soak them in water to cover by at least 2 inches for 12 hours. Make the stock: Place the ham hock, duck carcass, or ham bone, and the pancetta in a medium-sized saucepan. Add 6 cups water and bring to a boil, skimming. Add the garlic head and 1 teaspoon salt and a pinch of pepper. Reduce the heat to low and simmer, partially covered, for 1 hour. Make the fricassee: Melt the duck fat and the diced pancetta in a heavy 3-quart casserole set over moderately low heat. Add the sliced onions, leek, celery, and herb bouquet. Cover tightly and cook until the vegetables are soft but not browned, about 45 minutes. Strain the bone stock and set aside the pancetta and garlic. Pick meat off bones and set aside. Discard all the bones. Cool, cover, and refrigerate the stock and bits of meat. At the same time you are soaking the beans, make the hachis: Crush the cooked garlic to express the pulp, add to a food processor along with the reserved pancetta, the parsley, and piment d'Espelette. Puree until smooth. Scrape into a small bowl, cover, and refrigerate overnight. The following day, drain the beans and place in a deep, heavy saucepan, preferably copper. Degrease the bone stock and add to the saucepan along with 1 quart water; bring to a boil, skimming. Cover and cook the beans for 20 minutes over medium low heat. Add the leek and onion fricassee, the turnips, carrots, and potatoes, and simmer, uncovered, until beans and vegetables are tender, about 1 hour. In a large pot of boiling salted water, blanch quartered cabbage, uncovered, for 10 minutes. Drain and coarsely chop. Remove the pieces of confit from its softened fat; scrape off excess. Add to the saucepan along with the cabbage, the bits of meat, and the pureed hachis and bring to a boil over moderately high heat. Reduce the heat to moderately low and simmer slowly, uncovered, for 30 minutes. Cool, cover, and refrigerate for several days, up to 2 to 3 days. The garbure improves in flavor as it sits and mellows with time. Just before serving, check the consistency of the garbure; it should be so thick that a wooden spoon stands up straight in the center. If it is too thin, boil down. Stir in the remaining 1 tablespoon reserved confit fat, if desired. Season with additional salt, pepper, and piment d'Espelette to taste. To make a gratinéed variation, which is called a panade, layer slices of country-style bread, the boned duck, and the vegetables in a deep ovenproof or earthenware bowl. Moisten everything with the soup, sprinkle with grated cheese, such as Cantal or mild Cheddar, and set in a preheated 350°F, oven to bake until the top is crusty and brown, about 45 minutes.
