Make the No-Cook Tomato Sauce: Whisk all ingredients together in a bowl; set aside.
Make the No-Cook Cream Sauce: Whisk all ingredients together in a bowl; set aside.
Make the Lasagna Filling: Toss eggplant with 1 teaspoon salt in large bowl. Line surface of large plate with a double layer of coffee filters or paper towels and lightly spray with vegetable oil spray.
Spread eggplant in even layer over filters. Wipeout and reserve now-empty bowl. Microwave eggplant, uncovered, until dry to touch and slightly shriveled, about 10 minutes, tossing once halfway through to ensure that eggplant cooks evenly. Let cool slightly.
Return eggplant to bowl and toss with zucchini and squash.
Combine 1 T. oil, garlic, and thyme in small bowl. Heat 2 T. oil in a large skillet over medium-high heat until shimmering.
Add half eggplant mixture, salt, and pepper; cook, stirring occasionally until vegetables are lightly browned, about 7 minutes. Push vegetables to sides of skillet; add half of the garlic mixture to clearing and cook, mashing with spatula, until fragrant, about 30 seconds. Stir to combine garlic mixture with vegetables and transfer to medium bowl. Repeat with remaining eggplant mixture, 2 tablespoons oil, and remaining garlic mixture.
Return skillet to medium-high heat, add remaining teaspoon oil, and heat until shimmering. Add spinach and cook, stirring frequently, until wilted, about 3 minutes. Transfer spinach to a paper towel-lined plate and drain 2 minutes. Stir into eggplant mixture.
Spray 13 by a 9-inch baking dish with vegetable oil spray. Spread 1 cup tomato sauce in bottom of baking dish; shingle 4 noodles on top of sauce. Spread half of the vegetable mixture over noodles, followed by half of the olives, half of the cream sauce, and 1 cup of mozzarella. Repeat layering with 4 noodles, 1 cup tomato sauce, remaining vegetables, remaining olives, remaining cream sauce, and 1 cup mozzarella. Place remaining 4 noodles on the top layer of cheese. Spread remaining 1 cup tomato sauce over noodles and sprinkle with remaining 1 cup mozzarella.
Lightly spray large sheet of aluminum foil with vegetable oil spray and cover lasagna. Bake for 35 minutes at 375F on the middle rack of your oven, until bubbling. Cool on wire rack 25 minutes. Cut into pieces, sprinkle with basil, and serve.