Slice the zucchini lengthwise into 1/8-inch thick strips. Place in a bowl and salt generously, tossing once or twice to coat well. Lay strips on paper towels on your work surface. Set aside 1 hour.
Meanwhile, spray a large saucepan with nonstick spray and set over medium heat. Add the onion; cook, stirring often, until softened, about 2 minutes. Add garlic; cook 20 seconds.
Add the mushrooms; cook, stirring often, until they give off their liquid and it reduces to a glaze, about 7 minutes.
Stir in the tomatoes, basil, oregano, nutmeg and pepper. Cook, stirring occasionally, until the tomatoes start to break down and the sauce thickens, about 25 minutes.
Position the rack in the center of the oven and preheat the oven to 350°F.
Blot any moisture off the zucchini strips with paper towels. Use one third of the zucchini strips to line the bottom of a 9 x 13-inch baking pan, laying them lengthwise like you would lasagna noodles. Top evenly with one third of the sauce, then one third of the shredded cheese. Place half the remaining zucchini strips on top, as before, then top evenly with half the remaining sauce and half the remaining cheese. Repeat this process one more time: using the remaining zucchini, remaining sauce, and remaining cheese.
Bake, uncovered, until bubbling, about 45 minutes. Let stand at room temperature for 10 minutes before slicing into 6 pieces; serve.