Spaghetti with Tomato and Walnut Pesto
['2/3 cup walnuts', '2 pints cherry tomatoes, halved', '2 tablespoons plus 1/3 cup olive oil, plus more for drizzling', 'Kosher salt', '6 oil-packed anchovies, coarsely chopped', '2 garlic cloves, coarsely chopped', '1 teaspoon finely grated lemon zest', '1/4 teaspoon crushed red pepper flakes', '1/2 ounce Parmesan, finely grated (about 1/2 cup), plus more for serving', '1 teaspoon freshly ground black pepper', '12 ounces spaghetti', '1/2 cup (packed) basil leaves']
Preheat oven to 350°F. Toast walnuts on a rimmed baking sheet, tossing once, until slightly darkened, 8–10 minutes. Let cool.
Heat broiler. Toss tomatoes with 2 Tbsp. oil on a rimmed baking sheet; season with salt. Broil, tossing once, until tomatoes are blistered and have released some of their liquid, 5–7 minutes. Let cool.
Pulse anchovies, garlic, lemon zest, red pepper flakes, and 1/2 oz. Parmesan in a food processor until finely ground. Add walnuts and half of tomatoes, then, with motor running, stream in 1/3 cup oil; process just until combined. Season with salt. Transfer pesto to a large bowl and stir in black pepper.
Cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente. Drain, reserving 1/2 cup pasta cooking liquid.
Transfer pasta to bowl with pesto and add a splash of pasta cooking liquid. Toss, adding more cooking liquid as needed, until sauce coats pasta. Add basil and remaining tomatoes.
Divide among bowls; top with more Parmesan and black pepper and drizzle with oil.
Pesto can be made 1 day ahead. Cover and chill.
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