Put the dried beans in a large stock pot, adding three times as much water, and let them soak overnight.
The next day, drain and rinse the beans. Return them to the stock pot and add cold water to cover by 2 inches. Stick the cloves into the whole onion and add to the pot along with, carrots, bouquet garni, dried cured sausage slices, slab of bacon, 1 tablespoon tomato paste, and one teaspoon salt. Stir to mix the ingredients. Bring to a boil, and then reduce heat to a simmer. Cook the beans for about 1 to 1-1/2 hours. Beans should be cooked but still hold their shape.
In the mean time, in a large deep frying pan, heat 1 tablespoon of extra virgin olive oil over medium-high heat. Season the pork shoulder pieces with salt and pepper, add to the pan and brown the meat on all sides for about 5 minutes. Season the meat with piment d' Espelette and sprinkle with the thyme leaves. Add just enough water to barely cover the meat, bring to a boil and then turn heat down. Simmer for 30 minutes. Remove skillet from heat and scoop pork from liquid, reserving the liquid and pork separately.
In another large frying pan, heat 1 tablespoon of extra virgin olive oil over medium-high heat. Add the Toulouse/pork sausages and brown on all sides about 10 minutes. Turn the heat down to medium and continue to cook for 10 minutes. Reserve to a plate. Season the lamb pieces with salt and pepper. In the same pan, sear the lamb pieces until brown on all sides.
In a large Dutch oven, heat the remaining 2 tablespoons of extra virgin olive oil over medium-high heat. Add the sliced onions and garlic and sauté for about 5 minutes or until onions are golden. Retrieve the bacon slab from the bean pot and dice the bacon, and add to onions. Continue to cook for a few minutes.
Strain the beans, reserving the liquid and beans separately. Pick out the onion with cloves, bouquet garni and discard. Pick out the dried cured sausage and reserve. Add the beans to the Dutch oven, add the tomatoes, and remaining 1 tablespoon tomato paste. Gently stir to combine. Pour enough of the bean and pork shoulder liquid to just cover the bean mixture. Bring mixture to a boil over medium-high heat and then reduce heat. Simmer for 20 minutes.
Preheat oven to 350 F.
In the meantime, in a cast-iron skillet over medium-high heat, cook the duck legs skin side down until brown about 6-8 minutes, turn and cook another 6 minutes
Mix the bread crumbs with just enough extra virgin olive to moisten the bread crumbs.
Slice the Toulouse/pork sausages. Arrange the lamb, sausages, pork shoulder and duck legs on top of the beans. Sprinkle the bread crumbs over the meats. Drizzle a little olive oil over the top.
Bake in oven for 20-30 minute and then broil for 5-8 minutes until bread crumbs are golden. Serve immediately.
Rumor has it cassoulet is best reheated the next day. To reheat, remove the cassoulet from the refrigerator one hour before you plan to reheat. Preheat oven to 350 F. Break the bread crumb top to see if much liquid remains beneath the surface. If needed, add 1/2 cup warm water. Bake cassoulet for about one hour to reheat.