Sprinkle the meat on all sides with salt, then place it in a heavy, 8-quart pot. Place over medium-high heat and sear until browned on both sides, turning several times. Remove and set aside on a plate.
Immediately add the oil and the sliced onion. Saute for 5 minutes, until onions are wilted.
Add the tomato wedges and stir with the onions for 2 or 3 minutes, until the tomato juices start bubbling.
Add half the cabbage. Place the meat on the cabbage, then top with the remaining cabbage. Add water, salt, pepper and sugar. Cover and bring to a boil.
Lower heat and simmer, covered, very gently, for 2 to 2 1/2 hours, stirring occasionally, until the meat is so tender it practically falls apart when prodded with a fork.
Stir in the lemon juice and taste for seasoning, adjusting with additional salt, pepper, sugar or lemon juice, as desired.
Refrigerate the soup overnight, then skim off the hardened fat. Strip the meat into large pieces, discarding bones. Return the meat to the soup. Reheat the soup.
Serve piping hot in a deep bowl with a boiled potato on the side, or serve in a flat bowl with quarters or chunks of potato in the bowl with the soup. Sprinkle with dill only if you are one of those cooks who feel compelled to garnish; the mahogany colored soup is beautiful as is.
Advance Preparation: The soup is much better if made a day ahead and reheated. It also freezes very well. In either case, however, it's flavor will require refreshening. Taste carefully for salt, pepper and lemon juice.