In a large skillet over medium heat, toast the walnuts, stirring frequently, until fragrant, about three to five minutes. Transfer them to the bowl of a food processor and let them cool for a few minutes.
Add the arugula, Parmesan, garlic, 1 tablespoon lemon juice and salt. Process the mixture while drizzling in the olive oil, until it is mostly smooth.
Transfer the pesto to a bowl. Taste and add more lemon juice, for zing (up to 1 tablespoon), or salt, for overall flavor. Use as desired. Leftover pesto keeps well in the refrigerator, covered, for up to 1 week. Or, freeze it for up to 6 months. (I like to freeze pesto in ice cube trays. Once frozen, pop out the cubes and store in a freezer-safe bag.)