Heat oil in a soup pot over medium heat and add kielbasa. Cook, stirring, for a few minutes to flavor the oil. Add onions, celery, and carrots, and cook, stirring, until onions turn translucent, about 5 minutes.

Quarter potatoes and cut into 1/4-inch thick slices. Transfer about 1/3 of the potatoes into cold water, and reserve in the refrigerator until needed. Stir remaining sliced potatoes into the vegetables.

Add broth, water, salt, pepper, cayenne, pickles, pickle juice, and half the fresh chopped dill. Raise the heat to high, and wait for soup to start to simmer. While soup is coming up to temperature, stir in sour cream.

Reduce heat to medium-low, and simmer, stirring occasionally, until potatoes are tender, 40 to 45 minutes. Use a potato masher to break up potatoes.

Drain and add the reserved sliced potatoes, and let soup simmer until potatoes are very soft and tender, about 40 to 45 minutes more. During the last 30 minutes, be careful to stir gently to maintain whole pieces of potato.