Prep the cabbage: Bring a large pot of water to a simmer. Core the cabbage with a paring knife.
Lower the whole head into the water and simmer 8–10 minutes, peeling off softened outer leaves with tongs as they loosen. You’ll need about 10–12 large leaves. Drain leaves and pat dry.
Trim the ribs: Lay each leaf flat.
Use a knife to shave down the thick center rib so the leaf is flexible but still intact.
Mix the filling: In a bowl, combine ground meat, cooked rice, onion, garlic, egg, salt, pepper, smoked paprika, oregano, and parsley. Mix gently until just combined. Do not overwork.
Make the sauce: In a small bowl, stir together crushed tomatoes, tomato paste, sugar (if using), olive oil, and broth.
Season with a pinch of salt and pepper.
Assemble the rolls: Place 1/4 cup of filling near the base of a cabbage leaf. Fold the sides over, then roll up tightly like a burrito. Repeat with remaining leaves and filling.
Preheat the air fryer: Set to 360°F (182°C) for 3–4 minutes.
Lightly oil or spray the basket to prevent sticking.
Arrange and sauce: Place rolls seam-side down in the basket, leaving a little space between them. Spoon a bit of tomato sauce over each roll. Don’t drench them; reserve extra sauce.
Cook the first side: Air fry at 360°F (182°C) for 10 minutes.
Check halfway and rotate if your air fryer has hot spots.
Finish cooking: Flip the rolls carefully, spoon on a little more sauce, and air fry 8–10 more minutes, until the meat is cooked through and the edges are lightly browned. Internal temperature should reach 165°F (74°C).
Rest and serve: Let the rolls rest 5 minutes. Warm any remaining sauce and spoon over the top.
Serve with parsley, lemon, or a splash of vinegar for brightness.