Place basil leaves, pistachios, lemon zest, fresh lemon juice, and chopped garlic clove in the bowl of a food processor fitted with blade attachment. Process until very finely chopped.
With the motor running, drizzle in extra virgin olive oil.
Transfer to a bowl, then stir in grated Parmesan cheese. Add remaining olive oil if you want a thinner consistency, stirring until very well mixed. Taste, then adjust seasoning adding Kosher salt and freshly ground black pepper, as wanted.