Heat oil in a small skillet and cook onion for 5 minutes until softened. Add garlic for 30 seconds, then spread on a plate to cool; shave thick ribs from the cabbage leaves without cutting through them.
Mix raw beef, uncooked rice, cooled onion and garlic, filling water, sweet paprika and pepper. Divide into 12 portions and roll each inside a leaf with the sides tucked in, leaving room for the rice to expand.
Spread sauerkraut in the bottom of a wide heavy pot and arrange rolls seam-side down in two snug layers. Whisk broth, braising water, tomato paste and smoked paprika and pour over them; the liquid should come almost to the top layer.
Bring to a gentle simmer, cover and cook over low heat for 90 to 105 minutes. Check occasionally that liquid remains around the rolls and add the reserved extra water in 1/4-cup amounts only if needed to prevent drying.
Check that center fillings reach at least 160°F and the rice is tender, then rest covered off the heat for 10 minutes. Serve two rolls per person with sauerkraut and a little braising liquid.