Prepare the rice: Place the rice in a bowl and pour 3 cups of boiling water over it. Let it soak for 15 minutes, then drain well.
Sauté the vegetables: Heat the oil in a large skillet over medium heat. Add the onions and cook for about 3 minutes, until softened. Add the carrot and cook for another 3 minutes.
Add the tomato paste and butter: Stir in the butter and tomato paste. Cook for 3 to 4 minutes, stirring often, then transfer the mixture to a large bowl and let it cool slightly.
Mix the filling: Add the ground beef, ground pork, parsley, salt, black pepper, thyme, soaked rice, and egg. Mix gently just until evenly combined. Do not overmix.
Prepare the cabbage: Bring a large pot of water to a boil. Carefully remove the core from the cabbage, then lower the whole cabbage into the boiling water for 2 to 3 minutes, until the outer leaves begin to soften. Peel off the leaves as they loosen and set them aside to drain.
Prepare the leaves: Drain the cabbage leaves well. If the thick vein at the base feels tough, trim it slightly with a knife so the leaves are easier to roll.
Fill and roll: Place about 1 teaspoon of filling near the base of each cabbage leaf. Fold in the sides, then roll up snugly.
Arrange in the pot: Place the cabbage rolls seam-side down in a large pot, arranging them close together but not too tightly.
Make the sauce: In a small saucepan, combine the tomato puree, water, sour cream, sugar, and bay leaves. Bring to a gentle simmer and cook for about 3 minutes, stirring until smooth and well combined.
Cook the cabbage rolls: Pour the sauce over the rolls, making sure they are mostly covered. Cover the pot and simmer over low heat for 30 to 40 minutes, until the cabbage is tender and the filling is fully cooked.
Serve: Remove the bay leaves. Serve the stuffed cabbage rolls hot with plenty of tomato sauce, fresh parsley, and extra sour cream on the side if desired.