Cook whole head of cabbage in a large pot of boiling water for 2 minutes. Cool slightly. Remove softened outer leaves, keeping them whole; reserve leaves. Return cabbage head to pot, and repeat procedure to get 12 leaves. Finely chop remaining cabbage.
Heat a large skillet over medium-low heat. Add 1 tablespoon oil. Add onion; cook 5 minutes, stirring occasionally. Increase heat to medium-high. Add 2 cups chopped cabbage, carrot, and garlic; sauté 4 minutes. Remove from heat; cool slightly.
Combine remaining 2 tablespoons oil and strained tomatoes, stirring with a whisk. Spread 1 cup tomato mixture evenly in bottom of a 13 x 9-inch baking dish. Shave the rib of each cabbage leaf with a paring knife or vegetable peeler so it lays flat. Place 1/2 cup meat mixture in the center of each cabbage leaf. Starting at the core end, roll into a tight log, tucking in sides as you go. Place roll in baking dish, seam side down. Repeat with remaining leaves and filling. Top with remaining tomato mixture. Cover dish with foil; bake at 350° for 45 minutes. Uncover; shave remaining Parmesan cheese over top, and sprinkle with remaining 1 tablespoon dill.