Heat oil in a medium pot over medium-high heat. Break beef into 1/2-in. lumps and add to pot. Add onion and caraway seeds. Cook, stirring often, until meat is browned, about 15 minutes. Add cabbage and rice and cook, stirring often, until cabbage softens, 5 minutes. Stir in tomato paste to blend. Add 6 cups broth and bring to a boil. Add lemon juice and honey.
Reduce heat, cover pot, and simmer until rice is tender, 15 minutes. Remove from heat.* Stir in salt and chopped herbs.
Ladle soup into bowls and top with sour cream, dill sprigs, and parsley leaves.
*The soup thickens as it sits, so if you’re not sitting down right away, stir in more broth to thin it just before serving.