How to Make It
Place grated zucchini in a large bowl; squeeze with a paper towel to remove extra moisture. Heat oil in a large nonstick skillet over medium. Add shallots and garlic; cook, stirring occasionally, until softened, about 2 minutes. Add zucchini, pepper, and 1/2 teaspoon of the salt; cook, stirring occasionally, until vegetables have softened and volume reduces by about half, 18 to 20 minutes. (Mixture should be almost spreadable.)
Meanwhile, bring a large stockpot of salted water to a boil over high. Add pasta; prepare according to package directions. Drain pasta, reserving 1 cup cooking water.
Transfer pasta and reserved cooking water to zucchini mixture in skillet. Stir in butter and remaining 1/4 teaspoon salt. Cook over low, tossing occasionally, until mixture is well incorporated and sauce thickens slightly, about 2 minutes. Spoon pasta evenly into 4 bowls; garnish with basil leaves.