Cook the rice. Prepare the rice in a saucepan according to its package directions. Aim for approximately 2¼ cups cooked rice. Set aside while you finish the soup, or keep refrigerated if using previously cooked rice.
Brown the beef. Heat the olive oil in a 6- to 7-quart pot over medium-high heat. Add the ground beef and cook for 6 to 8 minutes, breaking it into small pieces. Let any released liquid cook off. Drain excess fat if needed. Cook the beef to 160°F, checked with a food thermometer.
Cook the vegetables. Add the onion and carrots. Cook for 5 to 7 minutes, stirring occasionally, until the onion starts to soften. Add the garlic, tomato paste, and paprika. Cook for 1 minute, stirring.
Build the soup. Add the cabbage, crushed tomatoes, tomato sauce, broth, Worcestershire sauce, bay leaf, salt, and pepper. Stir well and scrape any browned bits from the bottom of the pot.
Simmer. Bring to a boil, reduce the heat, and partially cover. Simmer gently for 25 to 35 minutes, stirring occasionally, until the cabbage and carrots are tender.
Add the rice. Remove the bay leaf. Stir in the cooked rice and simmer briefly until heated through. If using chilled leftover rice, heat the combined soup to 165°F. For meal prep, keep the rice separate and add properly heated rice to individual bowls instead.
Finish and serve. Stir in the vinegar, starting with ½ teaspoon if your tomatoes taste sharp. Taste and adjust the salt and pepper. Add more broth if needed, then serve with optional parsley or dill.