Make the fermented tomato sauce: Using a box grater, grate the vine-ripe tomatoes into a medium bowl, discarding the skins, and stir in the cherry tomatoes and salt. Cover the tomatoes with a square of cheesecloth and weight with a plate small enough to fit over the tomatoes but not touch the sides of the bowl. Place a heavy can or weight on the plate and let the tomatoes stand at room temperature. After 24 hours, remove the weights and rinse and wring out the cheesecloth. Replace the cheesecloth and weights over the tomatoes and repeat once a day for 4 more days. After the fifth day, remove the weights and discard the cheesecloth. Mash the cherry tomatoes into the sauce with a potato masher and scrape the tomatoes into a bottle or jar. Seal the jar and refrigerate the tomatoes until ready to use, or up to 2 weeks.
For the borscht: Roughly chop 2 carrots, halve 1 onion, and place both in a large saucepan along with the duck pieces, allspice, bay leaf, and 10 cups water. Season with salt and pepper, and bring to a boil. Reduce the heat to maintain a steady simmer and cook, skimming the foam from the surface occasionally, until the duck meat is falling off the bone, about 2 hours. Remove the pan from the heat, transfer the duck meat and bones to a bowl, and pour the stock through a fine strainer into a large bowl, discarding the solids. Let the stock and meat cool completely and then refrigerate both separately until chilled, at least 4 hours or overnight.
Remove the congealed duck fat from the surface of the chilled stock, place the fat in a large saucepan, and heat over medium. Cut the remaining carrot and onion into 1⁄4-inch dice, add to the pan, and cook, stirring, until soft and lightly caramelized, about 10 minutes. Add the bell pepper and cook, stirring, until soft, 5 minutes more. Pour the chilled stock into the pan along with the beet leaves, beet, and cabbage, and bring to a boil. Reduce the heat to maintain a steady simmer and cook, stirring, until the beet is just tender, about 10 minutes.
Shred the duck meat from the bones, stir into the soup along with the dill, and cook until warmed through, about 3 minutes more. Remove the soup from the heat and ladle into individual serving bowls. Serve the fermented tomato sauce on the side to stir into individual bowls of the borscht.