In a food processor fitted with a steel blade, combine the walnuts, pine nuts, and garlic. Process for about 30 seconds until finely chopped and well-mixed.
Add the basil leaves, kosher salt, and freshly ground black pepper. Pulse several times to thoroughly chop the basil, ensuring even distribution throughout the mixture.
With the processor running, slowly drizzle the olive oil into the mixture through the feed tube. Continue processing until the mixture is beautifully smooth and cohesive, but not over-blended into a paste.
Stop the machine and scrape down the sides. Add the freshly grated Parmesan and blend for another minute until emulsified and creamy.
Taste and adjust salt, pepper, or oil for desired consistency and flavor. If the pesto is too thick, add a little more olive oil.
Use the pesto immediately, or transfer to an airtight container. For storage, cover the pesto with a thin layer of olive oil to preserve freshness. Store in the refrigerator for up to a week or freeze for up to three months.
