Use a paring knife to cut out the core of the cabbage in a neat circle, and discard it, leaving the head whole.
Fill a large bowl with ice water, and place it by the stove. Bring a large pot of well-salted water to a rapid boil over high heat, then add the cabbage head, stem side up. Use tongs to carefully remove the leaves as they separate from the head, transferring them to the ice water as they release. When all the leaves are removed and chilled, drain them well and pat dry. Split the largest leaves in half, and cut away and discard any large, tough ribs.
In a medium bowl, whisk 2 cups cold water with the tomato paste and set aside. In a large bowl, combine the beef, ½ cup melted butter, the rice, 3 tablespoons cold water, ½ teaspoon salt, the cinnamon, and pepper. Use your hands to mix well.
Working one at a time, wrap ¼-cup portions of the meat mixture tightly in cabbage leaves to form a thick cigar shape, squeezing them snugly to secure. Arrange the rolls tightly along the bottom of a large Dutch oven in a circular pattern. Tuck the garlic cloves between the rolls. Sprinkle with the remaining teaspoon of salt, then add the tomato water. Place a plate on top of the rolls to keep them submerged. Cover the pot and bring the liquid to a boil over medium heat. Lower the heat to maintain a simmer, and cook until the rice is tender and the cabbage leaves are silky, about 45 minutes. Turn off the heat, uncover the pot, and drizzle the remaining butter over the top of the rolls. Cover and let stand off the heat for 15 minutes.
Place a large, rimmed platter upside down over the pot, and invert. Alternatively, use a large spoon to transfer the rolls and their juices to a rimmed platter and serve.