Pile up 2 or 3 layers of paper towels on a work surface. Working on the paper towels, shred zucchinis using a box grater holding the zucchini at an angle.
Heat a large skillet over moderate heat. Add extra-virgin olive oil to the pan Add chopped garlic to the oil. When garlic speaks by sizzling in oil, add shredded zucchini. Season zucchini with salt and pepper. Saute shredded zucchini 7 to 10 minutes. Add hot, drained pasta to the pan. Toss spaghetti with zucchini and garlic oil. Add a couple of handfuls of grated cheese to the pan. Adjust seasoning and serve.