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Heat the grill to medium. Put the chicken on a baking tray, drizzle with ½ tbsp oil and season. Grill for 8-10 mins on each side or until cooked through. Shred the meat with two forks.
Meanwhile, soak the bread in the milk for 5 mins, then tip into a food processor with the walnuts, remaining oil, garlic, lemon zest and juice. Pulse to a smooth creamy texture. Season.
Bring a large pan of salted water to the boil. Add the spaghetti and cook according to pack instructions.
Drain the spaghetti, reserving a mugful of cooking water (about 200ml). Return the spaghetti to the pan and toss with the walnut sauce, shredded chicken, most of the pasta water and the parsley. Toss to coat, and add a little more pasta water to loosen, if you need to. Check for seasoning and serve in bowls topped with the parmesan.