Detach and discard the first few outer leaves of the cabbage. The remaining head of leaves must be shredded very fine. If you are going to do it by hand, cut the leaves into fine shreds, slicing them off the whole head. Turn the head after you have sliced a section of it until gradually you expose the entire core, which must be discarded. If you want to use the food processor, cut the leaves off from the core in sections, discard the core and process the leaves through a shredding attachment.
Put the onion and olive oil into a large sauté pan, and turn the heat on to medium. Cook and stir the onion until it becomes colored a deep gold, then add the garlic. When you have cooked the garlic until it becomes colored a very pale gold, add the shredded cabbage. Turn the cabbage over 2 or 3 times to coat it well, and cook it until it is wilted.
Add salt, pepper, and the vinegar. Turn the cabbage over once completely, lower the heat to minimum, and cover the pan tightly. Cook for at least 1 1/2 hours, or until it is very tender, turning it from time to time. If while it is cooking, the liquid in the pan should become insufficient, add 2 tablespoons water as needed. When done, taste and correct for salt and pepper. Allow it to settle a few minutes off heat before serving. Note: The smothered cabbage can be prepared 2 or 3 days ahead of the soup, or served as a side dish from here. It also freezes well.
Rice and Smothered Cabbage Soup
Put the cabbage and broth into a soup pot, and turn on the heat to medium.
When the broth comes to a boil, add the rice. Cook uncovered, adjusting the heat so that the soup bubbles at a slow, but steady boil, stirring from time to time until the rice is done. It must be tender, but firm to the bite, and should take around 20 minutes. If while the rice is cooking, you find the soup becoming too thick, add a ladelful of homemade broth. If you are not using homemade broth, just add water. Remember that when finished, the soup should be rather dense, but there should still be some liquid.
When the rice is done, before turning off the heat, swirl in the butter and the grated Parmesan, stirring thoroughly. Taste and correct for salt, and add a few grindings of black pepper. Ladle the soup into individual bowls, and allow it to settle just a few minutes before serving. Serve with more grated Parmesan.
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