Rinse the cabbage and remove 8 large leaves. Cut out the thick center vein and blanch in boiling water. Remove and rinse immediately with cold water and drain. Cut the remaining cabbage into quarters lengthwise. Cut out the stalk and cut crosswise into strips. Cook about 7 minutes. Drain, rinse and drain again. Save the broth.
Soak the bread in water. Peel the onion and cut lengthwise into eighths. Chop the meat finely and mix with 3 tablespoons of cabbage, the drained bread, eggs, 2 tablespoons of ajvar, salt and pepper.
Spread open the cabbage leaves and distribute the filling. Pull up the sides of the leaves and fold into rolls. Tie with kitchen string.
Heat the oil in a roasting pan. Cook each roll on all sides. Add the onion and sauté briefly. Pour in the broth, cover and cook over medium heat for 30 minutes. Add the remaining ajvar, salt and pepper and cook for another 10 minutes.
Braise the remaining cabbage in butter for about 3 minutes. Season with salt and pepper.
Serve the rolls with the sauce, cabbage and boiled potatoes.