Combine beef shanks and water in a large stock pot. Bring to a boil and reduce heat to a very low simmer. Use a strainer to skim off any foam on the surface. Add carrot pieces, whole onions, celery stalks, bay leaf, and peppercorns. Simmer for 4 hours, skimming the surface occasionally.

Strain the stock. Discard the vegetables but reserve meat. Allow stock to cool to room temperature, then transfer to a covered container and refrigerate overnight. When beef shanks are cool, remove meat from the bones. Discard bones and gristle. Dice meat and any remaining marrow into small cubes. Cover the container and refrigerate overnight.

The next day, heat vegetable oil in a stock pot or Dutch oven over medium heat. Stir in garlic and 1 diced onion; cook and stir until onion has softened and turned translucent, about 5 minutes. Add paprika and cook 1 additional minute, stirring constantly. Gradually add cabbage and cook, stirring occasionally, until slightly wilted and reduced in bulk, about 8 minutes.

Remove beef stock from the refrigerator; skim and discard any fat. Pour stock into the pot; stir in reserved chopped beef. Add tomatoes with their juice. Bring soup to a boil, and then reduce the heat to a simmer. Cook the soup, partially covered, until cabbage is quite soft, about 1 hour.

Stir in lemon juice, sugar, dill, salt, and pepper. Add more water, if desired, for a thinner soup. Cook for 5 minutes, uncovered. Taste and adjust the seasonings before serving.