Preheat oven to 375°F. Line a large baking sheet with paper towels. Using a mandoline, shave zucchini lengthwise into very thin strips. Place about one-third of the strips in a single layer on paper towels; sprinkle evenly with about one-third of the salt. Top with another layer of paper towels; repeat process twice. Let stand 10 minutes; gently press strips with paper towels to remove excess moisture.

Rub inside of a 13- x 9-inch (3-quart) glass baking dish with olive oil; rub entire inside surface with cut sides of garlic.

Place 3 lasagna noodles in bottom of prepared dish. (They will not cover entire surface but will expand when cooked.) Top with 1 cup marinara sauce, spreading evenly to cover noodles. Top evenly with 1⁄4 cup ricotta, 1⁄4 cup Parmesan, and 2 ounces mozzarella. Cover with about one-fourth of the zucchini strips; sprinkle with about 4 teaspoons basil and 1⁄4 teaspoon pepper. Repeat layers twice. Layer with remaining 3 noodles, 1 cup marinara sauce, and 1⁄4 cup ricotta. Top with 1⁄4 cup Parmesan and remaining 2 ounces mozzarella. Arrange remaining zucchini strips on top in a lattice design. Sprinkle with remaining 1⁄4 cup Parmesan and 1⁄4 teaspoon pepper.

Cover with foil; bake in preheated oven 45 minutes. Remove foil; bake until top is lightly browned and noodles are tender, about 20 minutes more. Let stand at least 15 minutes before slicing.