In a Dutch oven (a heavy pot, usually made of cast iron, that you can use on the stovetop and in the oven), heat the oil over medium heat. Cook the sausage until well browned, breaking it up with a wooden spoon. Remove and drain on paper towels; set aside.
Pour out the excess oil from the Dutch oven. Add the chicken broth, vegetables, beans, thyme, salt, pepper, a third of the garlic, and the sausage and return to heat. Mix well, scraping up any brown bits that have stuck to the bottom of the Dutch oven. Bring to a boil. Reduce heat to low, cover, and simmer, stirring occasionally, about 1 hour, until thickened and the vegetables are tender.
Heat oven to 400° F. In a bowl, combine the breadcrumbs, parsley, butter, and remaining garlic. Sprinkle evenly over the cassoulet and place in the oven. Bake, uncovered, until the crust is golden brown, 10 to 15 minutes.
Keep it: Refrigerate leftovers in a smaller baking dish up to 2 days. Reheat at 375° F for about 20 minutes. To make ahead and freeze, prepare without the breadcrumb topping. Cover and freeze up to 4 months. When ready to cook, uncover the cassoulet, sprinkle with the breadcrumb mixture, and bake, unthawed, 45 minutes to 1 hour at 400° F.