Place the basil, toasted walnuts, sliced garlic, lemon zest, and lemon juice in a food processor fitted with blade attachment. Secure the lid, the pulse until very finely chopped.
With the motor running, slowly drizzle in the extra virgin olive oil, beginning with ½ cup, and increasing to ¾ cup, as needed. You want the sauce to be liquidy, with excess olive oil, as the cheese will help to thicken the sauce.
Transfer the sauce to a bowl, then stir in ½ cup grated Parmesan cheese. Taste and adjust seasoning, as needed.