Line a rimmed baking sheet with paper towels. Pour neutral oil and 1 cup olive oil into a Dutch oven and heat to 350°F. Fry ⅓ of zucchini rounds until golden brown, about 6-8 minutes, transferring to lined sheet.

Repeat frying remaining zucchini batches, sprinkling with ½ teaspoon kosher salt after cooking.

Bring a large pot of water to boil. Add remaining salt and spaghetti, cooking until softened but not al dente (about 3 minutes less than package instructions). Reserve 2 cups of cooking water, then drain.

In a skillet, heat remaining olive oil. Add garlic and cook until fragrant. Add half of the fried zucchini and mash into a paste with garlic. Add pasta, 1 ¼ cups reserved cooking water, and remaining salt, stirring until emulsified and spaghetti is al dente.

Adjust consistency with more pasta water as needed. Season with salt and pepper, then remove from heat.

Stir in grated cheese rapidly. Transfer to a serving bowl, garnishing with remaining fried zucchini and basil. Serve immediately.