Heat oil in a large pot or Dutch oven over medium heat. Add kielbasa and cook, stirring occasionally, until browned, 8 to 10 minutes. Use a slotted spoon to transfer to a plate. Add onion, carrot, and cabbage to pot. Season with salt and pepper. Cook, stirring occasionally, until vegetables are softened, 5 to 6 minutes. Add garlic and cook until fragrant, 1 to 2 minutes. Add broth, potatoes, and sausage. Bring to a boil, then reduce heat and simmer until potatoes are cooked through, 17 to 20 minutes.
Remove from heat. Stir in vinegar and dill. Serve with crusty bread.
