Put the onion and olive oil in a Dutch oven (or another pot of approximately the same size), and set over medium heat. Cook and stir until the onion is pale gold, and then add the garlic. Continue cooking until the garlic is fragrant and looks cooked through, a few minutes, and then add the sliced cabbage. Stir a few times to coat the cabbage with oil; then continue to cook until it’s wilted. Add a couple of generous pinches of salt, a grind or two of pepper, and the vinegar. Stir to mix, and then cover the pan and reduce the heat to the lowest setting. Cook, stirring occasionally, for at least 1.5 hours, or until the cabbage is very, very tender. If the pan seems dry at any point, you can add a tablespoon or two of water. When the cabbage is done, taste for salt, and season as needed.
This cabbage can be made a few days ahead of the soup, if needed, and it also freezes nicely.