Allrecipes.com has a very highly rated recipe for Ukrainian Red Borscht Soup, as do Food.com and NatashasKitchen.com. The recipes are all quite similar to each other and each claims to be authentic.
For Allrecipes.com's borscht, crumble 16 ounces of pork sausage into a skillet over medium-high heat and cook, stirring, until the pork is no longer pink, then set it aside. Bring to boil a large pot half-filled with about 2 quarts of water, add the sausage and cover. Return the water to a boil, then add 3 medium beets, peeled and shredded, and cook out their color. Add 3 medium carrots, peeled and shredded, and 3 medium baking potatoes, peeled and cubed. Cook for about 15 minutes, or until tender, then add half of a medium head of cabbage, cored and shredded, as well as an 8-ounce can of diced tomatoes, drained.
In a skillet over medium heat, warm 1 tablespoon of vegetable oil, then add 1 medium onion, chopped, and cook until tender. Add a 6-ounce can of tomato paste and 3/4 cup of water and stir until well-blended. Tip the contents of the skillet into the pot, then add 3 cloves of garlic, minced. Cover the pot and turn off the heat. Allow to stand for 5 minutes. Check seasoning and add salt, pepper or sugar if desired.
Serve the soup in bowls, garnished with sour cream and fresh parsley.