Pick over the beans, discarding any stones or other objects and any beans that are discolored. Place the beans in a deep bowl and pour in enough cold water to cover them by at least 4 inches. Soak the beans in a cool place at least 8 hours or overnight.

Drain the beans and transfer them to a heavy tall pot. Add 3 quarts of water, the potatoes, bay leaves and peperoncino. Bring to a boil over high heat. Adjust the level of heat to medium and cook, stirring occasionally, 30 minutes.

Meanwhile, drain the sauerkraut and taste it. If it is mild tasting, rinse it briefly in a colander under cold water and drain it. If the flavor is more assertive, rinse it twice before draining. In a medium-size skillet heat the olive oil over medium heat. Add the onions and cook, stirring often, until light golden, about 6 minutes. Stir in the trito and continue cooking until fragrant and very lightly browned, about 2 minutes. Add about 1 cup of the bean cooking liquid to the skillet and bring to a boil. Scrape the trito mixture into the pot. If using the ham hocks add them now. Continue simmering 30 minutes.

Add the sauerkraut to the pot. If using the pork butt, add it now. Cook, stirring and skimming occasionally, until the beans are tender, about 30 minutes. Meanwhile, transfer the potatoes to a bowl and mash them roughly. Remove the meat from the pot and cut into 1/4-inch dice; return the diced meat to the minestra. Return the potatoes to the pot for the last 10 minutes of cooking and season the soup with salt and pepper to taste. Remove the soup from the heat and let it stand 5 minutes. Serve the soup in warm bowls, sprinkled with the grated cheese.