Heat oil in an 8-quart stock pot or Dutch oven over medium-high. Add sausage and break it up with a wooden spoon into bite-size pieces. Let sausage pieces cook without stirring until brown on the bottom, 2 to 3 minutes. Using tongs, flip sausage pieces over and leave them until cooked through and golden brown, 2 to 3 minutes more.
Transfer sausage to a plate lined with paper towels, leaving the fat and oil behind in the pan. Cover sausage with aluminum foil to keep in any moisture; set aside.
Return pot to medium and add onions, oregano, fennel seeds, 1 teaspoon salt, and 1 teaspoon pepper. Cook, stirring often, until the onions are translucent and soft, 10 to 12 minutes. Add potatoes, stock, and, if desired, bay leaf. Cover and bring to boiling over high heat, then reduce to a simmer and cook until potatoes are fork-tender, 8 to 10 minutes.
Stir in beans, kale, and reserved sausage. Cook until beans are heated through. Fish out bay leaf, stir in vinegar, and season with salt and pepper.
Serve with a pile of freshly grated cheese on top.