Step 1

Bring 4 quarts water to boil in large pot.

Step 2

Meanwhile, process Parmesan in food processor until finely ground, about 30 seconds; transfer to medium bowl. Process parsley; oil; walnuts; capers; lemon juice; garlic; anchovies, if using; pepper flakes; salt; and pepper until smooth, about 1 minute, scraping down sides of bowl as needed. Transfer pesto to bowl with Parmesan and stir to combine. Season with salt and pepper to taste.

Step 3

Add pasta and 1 tablespoon salt to boiling water and cook, stirring often, until al dente. Reserve ¾ cup cooking water, then drain pasta and transfer to large bowl. Add pesto and ½ cup reserved cooking water to pasta and toss to combine. Season with salt to taste. Adjust consistency with remaining reserved cooking water as needed. Serve immediately, passing grated Parmesan separately.