Heat olive oil in a saucepan and soften the onions until they are translucent. Stir in the garlic, followed by the fire-roasted tomatoes, tomato sauce, and smoked paprika. Whisk in the brown sugar and apple cider vinegar. Let this sauce bubble gently on low heat for 15 minutes to concentrate the flavors.

Submerge the whole cabbage head in a large pot of boiling water. Cook for roughly 8-10 minutes until the outer leaves become pliable. Remove the head, run it under cool water, and carefully peel off 12 large leaves. Trim the thick, woody vein from the base of each leaf so they roll smoothly.

In a wide mixing bowl, combine the ground beef, cooked rice, beaten eggs, chopped onion, parsley, thyme, and cumin. Use your hands to thoroughly incorporate the spices into the meat and rice.

Portion the meat mixture into 12 equal logs. Place one log at the stem end of a cabbage leaf, fold the sides inward, and roll it forward tightly—much like a burrito.

Spread a thin layer of the prepared tomato sauce in a 9x13 baking dish. Arrange the cabbage bundles seam-side down to keep them from unraveling. Drench the rolls with the remaining sauce.

Cover the dish tightly with foil. Bake at 350°F for 90 minutes. This long, slow cook allows the cabbage to become buttery-soft and the meat to fully infuse with the tangy sauce.