Heat oil in a large Dutch oven. Add diced pork belly and cook until well browned and fat is rendered. Remove pork belly to a plate and set aside.
Generously season beef with salt and pepper and brown in rendered pork fat. Work in batches if necessary to avoid overcrowding the pan.
Return beef and pork belly to the Dutch oven. Add water and remaining broth ingredients except salt. Bring to a boil, then reduce heat and simmer for 2 to 3 hours until beef is tender and broth is developed to your liking. Add water as necessary to the pot to keep solids covered.
Once broth is done simmering, strain out solids. Measure the broth and add additional water as necessary to make 2 quarts. Add additional salt to taste.
Separate beef from the rest of the solids. Dice and set aside. Discard remaining solids.
Broth can be made a day ahead and refrigerated overnight.
Add beef broth, potatoes, bay leaf, and tomato paste. Bring to a boil and then simmer until potatoes are almost tender, about 15 minutes.
Add diced beef and green cabbage then simmer 5 to 7 minutes more until cabbage is just tender and not mushy.
Remove from heat and add lemon juice (or vinegar), sugar, salt, pepper and dill. Taste and adjust any seasonings to taste. Allow to rest for 5 to 10 minutes to allow flavors to blend.
Remove bay leaf. Serve with a dollop of sour cream and sprinkled with extra chopped dill.