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 turn golden, 4-5 minutes. Add tomato paste and cook, stirring often, until it turns a shade darker, about 2 minutes. Remove pas from heat and let cool 5 minutes.

Transfer walnut mixture (including the oil) to a food processor. Add chilies, lemon juice, 3 oz. Parmesan, and 1/2 teaspoon salt. Pulse until a thick paste forms.

Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until al dent'e. Drain pasta, reserving 1 1/2 cups of pasta cooking liquid.

Scrape pesto into a large bowl, and add butter, then pasta and 1/2 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.

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