Line a rimmed baking sheet with paper towels and set aside. Pour neutral oil and 1 cup olive oil into a pot. Heat to 350°F (175°C). Carefully fry ⅓ of the zucchini until light golden brown, 6 to 8 minutes. Transfer to the baking sheet, and repeat for remaining zucchini, seasoning with kosher salt.

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

In a skillet, heat 3 tablespoons olive oil over medium heat. Add garlic and cook until fragrant. Add half of the fried zucchini and mash into a paste. Add pasta and reserved water, and remaining salt. Cook, stirring until sauce is emulsified and spaghetti is al dente, 3 to 5 minutes.

Remove from heat, add grated cheese, and stir rapidly to combine. Adjust consistency with reserved pasta water. Season with salt and pepper. Transfer to a serving bowl and top with remaining fried zucchini and basil.