Cook the pasta according to the package directions. Drain and return it to the pot.
Meanwhile, in a food processor, pulse the walnuts and garlic until chopped.
Add the spinach, oil, ¼ cup of the Parmesan, and ¼ teaspoon each salt and pepper.
Puree until smooth, scraping down the sides of the processor bowl as necessary.
Add the pesto to the pasta and toss to combine. Sprinkle with the lemon zest and remaining ¼ cup Parmesan before serving.