Trim any tough ribs from the cabbage leaves and tightly roll 2 tablespoons of the meat mixture into each leaf (you should have 20 rolls). Shred remaining blanched cabbage (both the head and any remaining leaves).
Scatter half of the shredded cabbage leaves in a large Dutch oven. Arrange the cabbage rolls , seam-side down, on top of the shredded cabbage, fitting them snugly to prevent unrolling. (If you are using a smaller pot arrange the rolls in two layers). Sprinkle with remaining shredded cabbage over the rolls.
In a medium bowl, add diced tomatoes and juice, tomato puree, water, 3 tablespoons brown sugar, lemon juice, and 1/2 teaspoon of salt. Stir to combine, then pour over cabbage. Push the whole cloves into the whole, peeled onion and submerge it in the liquid. Bring the mixture to a boil, then reduce it to a simmer, cover, and cook for 30 minutes.
Discard the clove-studded onion. Sprinkle the crushed gingersnaps into pot and shake pot gently to incorporate them without disturbing cabbage rolls. Continue to simmer, uncovered, until the sauce begins to thicken, about 1 hour.
To serve, gently transfer the cabbage rolls to a serving dish using a slotted spoon. Stir in the remaining 3 tablespoons of brown sugar to the sauce and season to taste with salt and pepper. Spoon some sauce over the rolls and pass the extra sauce separately.