Soften the Cabbage Leaves
Core the cabbage and place it in a large pot of simmering water. Carefully remove 12 large leaves as they soften, about 3–4 minutes per layer. Set aside to cool and drain. Trim thick veins if needed for easier rolling.

Sauté the Aromatics
In a skillet, heat the vegetable oil over medium heat. Add the chopped onion and cook for about 5 minutes until softened. Stir in the garlic and cook for 1 more minute. Remove from heat and let cool slightly.

Make the Filling
In a large bowl, combine the ground beef, cooked rice, sautéed onion and garlic, 2 teaspoons of Hungarian paprika, salt, pepper, and beaten eggs. Mix until fully combined but not overworked.

Assemble the Rolls
Place a cabbage leaf on a flat surface. Add about 1/3 cup of the filling near the base. Fold the sides inward, then roll up tightly from the bottom. Repeat with the remaining leaves and filling.

Layer and Add Sauce
In a deep pot or Dutch oven, spread half of the sauerkraut on the bottom. Arrange the cabbage rolls seam-side down over the sauerkraut. In a separate bowl, whisk together water, tomato paste, and the remaining 2 teaspoons of paprika. Pour this mixture over the rolls and top with the remaining sauerkraut.

Simmer Gently
Cover the pot with a lid. Bring to a gentle simmer, then reduce the heat and cook on low for 1.5 to 2 hours. The cabbage should be tender and the filling fully cooked. Let the rolls rest for 10–15 minutes before serving.