Preheat the oven to 350°F. Add walnuts to a rimmed baking sheet and toast on the center rack for 8-10 minutes. Cook until slightly darkened, stirring halfway through. Allow to cool.
Heat broiler. In a large bowl, combine tomatoes, 2 tablespoons olive oil, and ½ teaspoon salt then add to a rimmed baking sheet. Place sheet in the oven about 8 to 10 inches away from the top of the broiler. Cook until tomatoes start to soften and blister, about 5 to 7 minutes. Set aside to slightly cool.
In a blender or food processor add anchovies, garlic, lemon zest, red pepper flakes, and ½ ounce of Parmesan cheese. Pulse or blend until finely ground, about 15 seconds.
Add ⅔ cup of the toasted walnuts and half of the tomatoes. With the motor running, stream in ⅓ cup of the olive oil, process until just combined. Taste and season with salt and pepper. Puree for a few more seconds until combined. Transfer pesto to a large bowl.
Cook pasta in a large pot of boiling salted water, according to the manufacturer's directions. Reserve ½ cup of the pasta liquid, then drain the noodles.
Add pasta to the pesto. Add a small amount of the pasta liquid and toss, add more liquid as needed until the sauce coats the pasta. Add basil and remaining tomatoes on top.
Divide among bowls and top with parmesan, chopped walnuts, basil, black pepper, and a little olive oil drizzle if desired.