Step 1: Toast the walnuts in a dry skillet over medium heat for 3-5 minutes until fragrant and slightly golden. Let them cool completely before proceeding.
Step 2: In a food processor or high-speed blender, combine the cooled walnuts, peeled garlic cloves, sea salt, and black pepper. Pulse until the ingredients are coarsely chopped.
Step 3: Add the fresh basil leaves and lemon juice to the food processor. Pulse several times until the basil is finely chopped and incorporated with the nut mixture.
Step 4: With the motor running on low speed, slowly drizzle in the extra-virgin olive oil in a steady stream until the pesto is smooth and emulsified.
Step 5: Add the grated Parmesan cheese and pulse a few more times just until combined. Do not over-process at this stage to maintain a bit of texture.
Step 6: Taste and adjust seasoning with more salt or lemon juice if desired. Transfer to a glass jar and serve immediately or store in the refrigerator.