Dice the onion, mince the garlic, and chop the cored cabbage into spoon-sized pieces (about 5 cups). Open the canned tomatoes, sauce, and paste, and measure out the spices.
Heat the olive oil in a Dutch oven over medium heat. Add the ground beef, onion, salt, and pepper, and cook for 8 to 9 minutes, breaking the beef into crumbles, until no pink remains. Spoon off excess fat if needed.
Stir in the garlic and cook for 1 minute until fragrant, then add all the cabbage. Cook for 5 minutes, stirring often, until the cabbage has softened slightly and begun to wilt.
Pour in the broth, diced tomatoes with juices, and tomato sauce. Add the tomato paste, Worcestershire sauce, brown sugar, paprika, thyme, oregano, and bay leaf. Stir well, scraping up browned bits, and bring to a boil, about 5 minutes.
Reduce heat to medium-low and stir in the uncooked rice. Cover and simmer for 30 to 35 minutes, stirring every 10 minutes, until the rice is tender and the cabbage is completely soft.
Remove from heat and discard the bay leaf. Stir in the apple cider vinegar, then taste and adjust salt, pepper, or brown sugar. Thin with up to 1 cup of extra broth if desired.
Ladle into bowls and top with chopped fresh parsley. If using sour cream, add about 1 tablespoon to each bowl off the heat, stirring in or leaving as a dollop.