Bring a large pot of salted water to a boil over high heat. Meanwhile, make the pesto sauce by blending or processing 8 Tbsp. olive oil along with the garlic, basil, walnuts, lemon and cheese until smooth. Season with salt and pepper to taste; set aside.
Add the pasta to the rapidly boiling water and cook until al dente, about 4-5 minutes. Strain well, reserving 1 cup cooking liquid. Return the pasta and cooking liquid to the large pot over a very low flame. Add pesto and stir gently until combined, seasoning with salt and pepper, to taste. Divide the pasta between 4 plates and serve with more grated cheese, if desired.