Prep the cabbage: Bring a large pot of water to a low boil. Core the cabbage.
Gently peel off 12–14 leaves, or blanch the whole head for 3–5 minutes and then remove leaves. Trim the thick rib on each leaf so it folds easily.
Make the sauce: In a bowl, whisk together tomato sauce, crushed tomatoes, tomato paste, brown sugar, vinegar, and beef broth. Taste and adjust for tang and sweetness.
You want a lightly sweet, tangy sauce.
Mix the filling: In a large bowl, combine ground meat, onion, garlic, cooked rice, egg, parsley, oregano, paprika, salt, pepper, and red pepper flakes if using. Stir just until mixed. Don’t overwork it.
Assemble the rolls: Lay a cabbage leaf flat, rib side down.
Spoon 1/4 cup of filling near the base. Fold the sides over the filling, then roll up tightly from the base to the tip. Repeat with remaining leaves and filling.
Layer in the slow cooker: Drizzle olive oil in the bottom of the slow cooker.
Spoon a thin layer of sauce to coat the bottom. Arrange rolls seam-side down in a single layer. Add more sauce, then a second layer of rolls if needed.
Pour remaining sauce over the top.
Cook: Cover and cook on Low for 6–7 hours or High for 3–4 hours, until the cabbage is very tender and the filling is cooked through.
Rest and serve: Let the rolls rest 10 minutes so the sauce thickens slightly. Spoon extra sauce over the top. Garnish with dill or parsley.
Add a dollop of sour cream if you like.