1Heat the oil in a large, deep skillet over medium-high heat. Add the beef, season with a good pinch of salt and pepper, and break it up as it browns, about 6 minutes. Leave the browned bits in the pan.
2Add the onion and cook, stirring, until soft and golden at the edges, about 5 minutes. Stir in the garlic, tomato paste, paprika, and marjoram and cook for 1 minute, until the paste darkens a shade.
3Stir in the cabbage in handfuls, letting each batch wilt a little before adding the next, until it all fits in the pan, about 4 minutes.
4Add the crushed tomatoes, broth, rice, brown sugar, and vinegar, and tuck in the bay leaves. Stir well, scraping up the bottom of the pan, and bring to a simmer.
5Cover, reduce the heat to low, and simmer for 25 to 30 minutes, stirring once or twice, until the rice is tender and the cabbage is soft. If the pan looks dry before the rice is done, add a splash of water.
6Remove and discard the bay leaves. Taste and adjust the salt, pepper, and vinegar to your liking.
7Serve in bowls topped with a spoonful of sour cream and a sprinkle of fresh dill.