Begin by toasting the walnuts in a dry skillet over medium heat for 3-4 minutes, or until they are fragrant and lightly browned. Stir frequently to prevent burning. Remove from heat and allow to cool.
Wash the basil and mint leaves, then pat them dry with a clean kitchen towel or paper towels to remove any excess moisture.
Peel the garlic cloves.
In a food processor, combine the toasted walnuts and garlic. Pulse until they are finely chopped.
Add the fresh basil and mint leaves to the food processor. Pulse a few times to break down the leaves.
While the food processor is running, slowly pour in the extra virgin olive oil. Process until the mixture is well combined and reaches your desired consistency. For a smoother pesto, process for a longer period.
Add the grated Parmigiano-Reggiano cheese, lemon juice, salt, and black pepper. Pulse until all ingredients are evenly incorporated into the pesto.
Taste the pesto and adjust the seasoning if necessary.
If not using immediately, transfer the pesto to an airtight container. Press a piece of plastic wrap directly onto the surface of the pesto to prevent oxidation, then seal the container. Refrigerate until ready to use.