Preheat oven to 350°F. Toast hazelnuts on a rimmed baking sheet, tossing once, until golden brown, 8–10 minutes. Let cool, then coarsely chop.
Place cherry tomatoes in a large bowl; season with salt.
Cook spaghetti in a large pot of boiling salted water, stirring occasionally, until al dente. Drain pasta, reserving 1/4 cup pasta cooking liquid.
Meanwhile, puree beefsteak tomato, garlic, red pepper flakes, 1/2 cup basil, 3 tablespoons chopped hazelnuts, and 1 teaspoon salt in a food processor until smooth; add to bowl with salted cherry tomatoes. Add zucchini, spaghetti, pasta cooking liquid, 1/4 cup oil, and remaining 1/2 cup basil leaves and toss to combine; season with salt and pepper.
Divide pasta among bowls, drizzle with more oil, and top with ricotta salata and remaining hazelnuts.