Heat the oil in a large Dutch oven over medium heat. Season the duck pieces with salt and black pepper. Once the oil is hot, sear the duck in the pan, fat side down for 7 to 8 minutes on the first side, and 5 to 6 minutes on the other side. Remove the duck from the pan and set aside on a platter. Discard all but 1 tablespoon of the fat from the pan.

Add the sausage and cook, stirring occasionally, until the sausage has rendered some of its fat and is slightly caramelized, about 5 to 6 minutes. Add the onions and celery to the pan and continue to cook, scraping the bottom of the pan, until the onions are translucent, about 5 minutes. Add the garlic, navy beans and bay leaves to the pan and cook, stirring, for 2 to 3 minutes. Pour the duck stock and water to the pan, and add the thyme as well as the reserved duck pieces back.

Bring to a boil then reduce to a simmer. Cook the soup, covered, for 2 hours, stirring occasionally and skimming any fat or scum that may rise to the surface. After 2 hours, test to see if the meat and beans are both tender, if not continue to cook uncovered.  Season the soup with the cayenne and remaining salt. Garnish with the greens onions and parsley, and serve while hot.