Bring a large pot of salted water to a boil, remove the core from the cabbage head, and boil the whole head for 5-8 minutes until the outer leaves soften and peel away easily.

Remove the softened leaves and set them aside to drain on paper towels; use a small knife to shave down the thickest part of the central vein on each leaf to make rolling easier.

In a large mixing bowl, combine the ground beef, cooked rice, chopped onion, minced garlic, beaten egg, dried thyme, 1 teaspoon of salt, and black pepper. Mix gently with your hands until the ingredients are just combined.

Place approximately 1/3 cup of the meat mixture at the stem end of a prepared cabbage leaf. Fold the sides in over the filling and roll tightly from the bottom up, ensuring the roll is secure.

In a large Dutch oven or deep pot, whisk together the crushed tomatoes, brown sugar, apple cider vinegar, beef broth, and the remaining 0.5 teaspoon of salt to create the braising sauce.

Arrange the cabbage rolls seam-side down in the pot, packing them snugly into the sauce; if needed, stack them in a second layer.

Bring the sauce to a gentle simmer over medium heat, then reduce the heat to low, cover the pot tightly with a lid, and cook for 60 to 75 minutes.

Test for doneness by piercing a roll with a knife to ensure the cabbage is tender and the filling is cooked through to 160°F.

Serve the warm cabbage rolls with a generous ladle of the tomato sauce over the top and a slice of rustic bread.