Step 1: Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium heat. Add chopped onion and cook until softened and translucent, about 5-7 minutes. Stir in minced garlic and cook for another minute until fragrant.
Step 2: Add ground beef to the pot. Break it up with a spoon and cook until fully browned and no longer pink. Drain any excess fat, leaving just a little for flavor.
Step 3: Stir in tomato paste and cook for 2-3 minutes, stirring constantly. Add Worcestershire sauce, dried marjoram, smoked paprika, salt, and black pepper. Stir well to coat the meat and aromatics with the spices.
Step 4: Pour in beef broth, crushed tomatoes, and diced tomatoes with their juice. Use a wooden spoon to scrape up any browned bits from the bottom of the pot. Stir in apple cider vinegar and brown sugar.
Step 5: Stir in shredded cabbage and uncooked rice. Bring the soup to a simmer, then reduce heat to low, cover the pot, and let it cook for 20-25 minutes, or until the rice is tender and the cabbage is soft. Stir occasionally to prevent rice from sticking.
Step 6: Taste the soup and adjust seasonings as needed. Add more salt, pepper, sugar, or vinegar to achieve your desired sweet and sour balance. If the soup is too thick, add a little more broth or water.
Step 7: Ladle the hot soup into bowls. Garnish with a dollop of sour cream or Greek yogurt and a sprinkle of fresh chopped parsley or dill, if desired. Serve immediately.