Toast the walnuts in the oven at 350°F for 8 minutes or on the stove over medium heat. You just want them very lightly toasted and fragrant. Let them cool.
Pulse the arugula leaves, cooled walnuts, and garlic cloves until finely chopped in a food processor. Add the shredded cheese, lemon zest, and lemon juice and pulse again a few times.
Turn the processor on and add the EVOO in a steady stream. Continue until you make a smooth paste. Scraping the sides is necessary.
Season with salt and pepper and store in the fridge. Should keep a few weeks. If you decide to freeze it, don’t add the cheese. Add it after you thaw it.