Very thinly slice zucchini crosswise (about 1/16 inch thick) with a mandoline or other manual slicer.
Heat 1 tablespoon butter in a 12-inch nonstick skillet over moderately high heat until foam subsides, then sauté zucchini and garlic, stirring, until zucchini is just beginning to soften, about 8 minutes. Transfer to a bowl and cool 10 minutes. Add basil and salt and pepper to taste.
Melt remaining tablespoon butter in an 8-inch nonstick skillet and remove from heat. Arrange one fourth of zucchini mixture in a layer in skillet and sprinkle with one fourth of parmesan. Repeat layering 3 times.
Cook skillet cake over moderate heat until bottom of cake is well browned, 5 to 7 minutes. Slide cake onto a large flat plate, then place skillet over cake and, holding plate firmly, invert cake back into skillet. Cook cake, shaking skillet frequently to prevent sticking, until underside is browned, 4 to 5 minutes more.
