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Bring medium pot salted water to boil.
In food processor or blender, combine walnuts, garlic, arugula, Parmesan, lemon juice and 1 tsp salt; pulse to blend. With the machine running, pour in olive oil through the food tube in a slow, steady stream and process until smooth, stopping to scrape the sides of the bowl as needed. Taste and adjust seasonings.
Add spaghetti to boiling water and cook according to package directions. Drain pasta, reserving a ¼ cup of pasta water. Toss spaghetti with arugula pesto. Add some of the reserved pasta water to loosen the sauce a bit, if desired. Divide between two plates and top with more grated cheese.