Line a rimmed baking sheet with a triple layer of paper towels and set aside. Pour neutral oil and 1 cup olive oil into a large Dutch oven or large heavy-bottomed pot. Heat oil over medium heat to 350°F (175°C). Carefully place ⅓ of the zucchini rounds into oil and fry until light golden brown, 6 to 8 minutes, stirring often. Use a spider skimmer to transfer fried zucchini to the lined baking sheet. Return oil to 350°F and repeat frying remaining batches of zucchini. Sprinkle fried zucchini with ½ teaspoon kosher salt. Let cooking oil cool completely before disposing or storing.

Bring a large pot of water to a boil over high heat. Add remaining 2 tablespoons salt, return to a boil, then add spaghetti and cook until pasta is softened on the exterior, about 3 minutes less than package directions. Reserve 2 cups of cooking water, then drain pasta.

In a large high-sided skillet, heat remaining 3 tablespoons olive oil over medium heat until shimmering. Add garlic and cook, stirring often, for about 1 minute. Add half of the fried zucchini and mash it into a paste with the garlic. Add par-cooked pasta and 1 ¼ cups reserved cooking water, and remaining 1 teaspoon salt. Cook, stirring vigorously, until sauce is emulsified and spaghetti is al dente, about 3 to 5 minutes.

Remove skillet from heat, add the grated cheese, and stir rapidly to combine. Adjust sauce consistency as desired with more pasta water. Season to taste with salt and pepper. Transfer to a serving bowl and arrange remaining fried zucchini rounds and basil over top. Serve.