Bring a large pot of water to a boil. Cook cabbage in boiling water until outer leaves pull away easily from the head, 3 to 5 minutes. Separate leaves and let cool. Reserve 1 1/2 cup cooking water.
Heat oil in a large skillet. Add onion and garlic; cook and stir until translucent, about 5 minutes. Stir in wet rice; cook and stir until moisture evaporates, about 1 minute. Remove from heat and let cool, about 5 minutes.
Mix beef, sausage, and egg together in a large bowl. Stir in cooled rice mixture.
Place a golfball-sized amount of the beef mixture on the ribbed end of a cabbage leaf; roll up, tucking in the sides. Repeat with remaining beef mixture and cabbage leaves.
Place rolls seam-side down in a slow cooker. Cover with tomato juice. Pour reserved cooking water on top.
Cook on High for 30 minutes. Switch to Low and cook for 6 1/2 hours.
Substitute vegetable juice for the tomato juice if desired.
If cooking on the stove, boil in a large pot for 30 to 45 minutes.
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