Slice cabbage into thin slices by hand or by food processor. Peel and chop onion into small chunks. Wash and dry apple, then grate.
In a pot, heat olive oil, then add sugar and gently stir until it foams. Then, add chopped onions and fry until translucent. Add sliced cabbage and red wine and mix well. Let it cook for a couple of minutes on medium heat, until the wine has reduced. Then, add grated apples and vegetable stock, season to taste and let it cook on low with the lid on for about 40-60 minutes, until the cabbage gets soft. Stir occasionally. Cooking time depends on the thickness of your slices - the thinner the cabbage is sliced, the sooner it will cook. About 10 minutes before the dish is done, add balsamic vinegar, mix well and continue cooking.