Put a large saucepan of water on to boil, and toast the walnuts in a dry frying pan until they begin to make a nutty aroma. Put the bread in a bowl and cover with the milk. Put most of the toasted walnuts (reserving about 1 ounce for garnish) into a blender along with the garlic, bread soaked in milk, and Parmesan. Blend until it turns smooth and creamy, then pour in the oil and season well with salt and pepper before blending again.
Pour into a bowl, and set aside.
Add pasta to boiling water with salt and cook for the required amount of time. When pasta is al dente, reserve a cup of the pasta cooking liquid and then drain the pasta, but put it in to a large bowl while it's still dripping slightly with water. Sprinkle a little olive oil over the pasta to prevent it sticking together, and then add the walnut sauce, mixing it into the pasta (splash in a little pasta cooking liquid to make the sauce less thick if needed).
Roughly chop the remaining walnuts and toss them over the top along with some more Parmesan and the chopped parsley.