Julienne skin (only) from half of zucchini with slicer; toss with 1/2 teaspoon salt and drain in a sieve until wilted, at least 20 minutes. Coarsely chop remaining zucchini.
Cook onion and garlic in oil in a 3- to 4-quarts heavy saucepan over medium-low heat, stirring occasionally, until softened, about 5 minutes. Add chopped zucchini and 1 teaspoon salt and cook, stirring occasionally, 5 minutes. Add 3 cups water and simmer, partially covered, until tender, about 15 minutes. Purée soup with basil in 2 batches in a blender (use caution when blending hot liquids).
Bring remaining cup water to a boil in a small saucepan and blanch julienned zucchini 1 minute. Drain in a sieve set over a bowl (use liquid to thin soup if necessary).
Season soup with salt and pepper. Serve in shallow bowls with julienned zucchini mounded on top.