Heat olive oil in a large Dutch oven over medium-high heat. Add ground beef, break it apart, and cook 5 to 7 minutes until no pink remains. Drain excess fat, leaving about one tablespoon.
Reduce heat to medium. Add diced onion and cook 3 to 4 minutes until softened. Add minced garlic and stir 30 seconds until fragrant.
Stir in tomato paste and cook directly in the pot for 1 minute, stirring constantly, to deepen the flavor.
Pour in beef broth, diced tomatoes with their juices, and tomato sauce. Add Worcestershire sauce, smoked paprika, thyme, salt, pepper, and brown sugar if using. Stir well to combine.
Add the chopped cabbage to the pot and stir to coat it in the broth mixture. It wilts down quickly as the liquid heats.
Bring the soup to a gentle boil, reduce to a low simmer, then stir in the uncooked rice. Cover and cook 20 to 25 minutes, stirring every 8 to 10 minutes to prevent sticking.
When the rice is tender and the cabbage is fully softened, taste and adjust salt and seasoning. Add extra broth if the soup thickens too much.
Ladle into deep bowls and top with a spoonful of sour cream and fresh dill or parsley if desired. Serve hot.
