Using a mandoline, slice the zucchini into uniform 2mm discs. Heat the extra virgin olive oil in a deep skillet to 170°C (340°F).
Fry the zucchini in batches until they achieve a deep golden-brown color. Remove with a slotted spoon and drain on paper towels. Season lightly with salt while hot.
Optional Technical Step: Allow the fried zucchini to rest in a bowl for at least 1 hour (or refrigerated overnight) to soften and develop a concentrated flavor profile.
Boil a large pot of water. Add salt until it tastes like the sea. Cook the spaghetti until it is 2 minutes shy of al dente.
While the pasta cooks, place a large wide skillet over medium heat with 2 tablespoons of fresh olive oil. Sauté the crushed garlic until golden, then discard the garlic cloves.
Add the rested zucchini to the skillet along with two ladles of starchy pasta cooking water. Use a wooden spoon to slightly break up some of the zucchini to help create a base for the emulsion.
Transfer the undercooked spaghetti directly into the zucchini skillet using tongs. Toss vigorously over medium-high heat, adding more pasta water as needed, until the pasta is perfectly al dente and coated in a light zucchini-infused broth.
Remove the skillet from the heat source entirely. This is critical to prevent the cheese from clumping.
Add the butter, grated cheese, and half of the basil. Stir and toss rapidly (the 'mantecatura' process), adding splashes of pasta water as necessary until a thick, creamy sauce forms and clings to the noodles.
Garnish with the remaining basil and a generous amount of black pepper. Serve immediately on warmed plates.