Prepare the rice: Pour 3 cups of boiling hot water over 1 cup of rice and let it sit for 15 minutes. Drain any excess water.
Saute the vegetables: Finely dice 2 onions and 1 carrot. Heat 4 tablespoons of oil in a pan over medium heat, add the onions, and cook for 3 minutes. Add the carrot and cook for another 3 minutes.
Add tomato paste and butter: Stir in 1 tablespoon of butter and 1 tablespoon of tomato paste. Cook for 3–4 minutes, then transfer the mixture to a large bowl.
Mix the filling: Add ground beef, ground pork, parsley, salt, black pepper, thyme, the soaked rice, and 1 egg. Mix everything until well combined.
Prepare the cabbage: Bring a large pot of water to a boil. Add the whole cabbage and cook for 2–3 minutes. Carefully separate the leaves and trim the thick core if needed.
Fill the cabbage leaves: Add about 1 teaspoon of filling to each cabbage leaf. Roll tightly, tucking in the sides as you go.
Arrange in the pot: Place the cabbage rolls seam-side down in a pot, layering them snugly but not too tight.
Make the sauce: In a bowl, mix tomato puree, water, sour cream, sugar, and bay leaves. Simmer for 3 minutes.
Cook the cabbage rolls: Pour the sauce over the rolls, cover the pot with a lid, and cook on low heat for 30–40 minutes until tender.
Serve: Remove bay leaves and serve warm with extra sour cream and freshly chopped parsley.