Step 1

In a large pot of boiling salted water, cook cabbage, turning often, until tender, 15 to 20 minutes. Drain and, with tongs, gently separate leaves. Cut out thick ribs from 8 to 10 of the largest leaves and set leaves aside. Shred remaining cabbage and place in an 8-inch square baking dish set on a baking sheet.

Step 2

Preheat oven to 375 degrees. In a medium saucepan, heat oil over medium-high. Add onion and cook until soft, about 5 minutes. Add garlic and stir until fragrant, 1 minute. Transfer half the onion mixture to a large bowl. Add tomatoes, sugar, and vinegar to saucepan, season with salt and pepper, and bring mixture to a boil. Reduce heat and simmer 10 minutes. Spread 1 cup tomato sauce on top of shredded cabbage in dish.

Step 3

To bowl with onion mixture, add rice, ground beef, and egg and season with salt and pepper. With your hands, mix well, then divide into 8 portions. Working with a few leaves at a time, lay cabbage on a work surface, overlapping smaller leaves to form one larger leaf if necessary. Place a portion of beef in the center of each. Fold bottom half of cabbage over filling, then fold in sides and roll up tightly. Transfer cabbage rolls, seam side down, to baking dish. Top with remaining sauce, cover dish with foil, and bake on sheet 40 minutes. Remove foil and bake until sauce is bubbling, about 10 minutes.