Step 1Season the short ribs with a teaspoon of salt.
Step 2Heat a 5-quart heavy-bottom pot over high heat until hot. Add the olive oil, then the short ribs in batches and heat until browned on each side, about 10 minutes. Remove the ribs from the pot and set aside.
Step 3Reduce the heat to medium-high and stir in the onions and garlic. Cook, stirring occasionally and scraping the browned bits from the bottom of the pot, until the onions are soft and translucent, 8 to 10 minutes.
Step 4Add the ribs back to the pot with the peppercorns and bay leaves. Add water to cover and bring to a simmer. Reduce the heat to a simmer, loosely cover and cook until the ribs start to become tender, about 1 1/2 hours.
Step 5Skim any foam or fat from the surface and stir in the turnips and carrots. Cook until the vegetables are soft and the ribs are very tender and falling off the bone, 30 to 45 minutes more.
Step 6Remove the ribs from the pot and discard the bones. Slice or pull the meat into coarse, medium chunks. Return the meat to the pot.
Step 7Stir in the tomato juice, kvass and cabbage and bring the mixture to a gentle simmer. Using a box grater or food processor, shred the beets into fine strips and add to the pot. Add the vinegar.
Step 8While the soup is simmering, prepare the dumplings: In a small bowl, whisk together the flour, baking powder, salt and caraway seeds. In a separate small bowl, beat together the milk and egg. Drizzle the egg mixture over the flour and stir with a fork until it forms a stiff, sticky batter.
Step 9Taste the soup and season as desired with salt and pepper, then make the dumplings: Using a tablespoon, drop spoonfuls of the batter into the simmering soup. Cover the pot and steam the dumplings until they are set and firm and the cabbage is tender, 10 minutes. Season to taste with salt and pepper. Spoon into bowls, making sure each serving gets at least one dumpling.