Step 1

Cook anchovies, garlic, oil, and walnuts in a small saucepan over medium heat, stirring often, until walnuts are deeply golden and garlic is just beginning to turn golden, 4–5 minutes. Add tomato paste and cook, stirring often, until it turns a shade darker, about 2 minutes. Remove pan from heat and let cool 5 minutes.

Step 2

Transfer walnut mixture (including the oil) to a food processor. Add chiles, lemon juice, 3 oz. Parmesan, and ½ tsp. salt. Pulse until a thick paste forms.

Step 3

Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente. Drain pasta, reserving 1½ cups pasta cooking liquid.

Step 4

Scrape pesto into a large bowl and add butter, then pasta and ½ cup pasta cooking liquid. Using tongs, toss pasta vigorously, adding more pasta cooking liquid if needed (you may not need all of it), until glossy and well coated with sauce.

Step 5

Divide pasta among bowls. Season with more salt and top with Parmesan.

Step 6

Do Ahead: Pesto can be made 5 days ahead. Cover and chill.