Heat 2 tbsp of olive oil in a large saucepan, add the onions and cook over a low heat until softened. Add
half of the shredded red cabbage and all the sugar and vinegar, and cook over a low heat for 15-25 minutes, or until softened.
Bring a large pan of salted water to the boil, add the remaining red cabbage and cook for 4 minutes. Add the pappardelle and cook for a further minute. Reserve 100ml (3½fl oz) of cooking liquid and then drain.
Toss the pasta and cabbage with the braised cabbage and reserved cooking liquor. Season to taste. Divide between 4 plates and top with Parmesan, walnuts and chopped parsley. Drizzle with olive oil.