Toss the eggplant with a teaspoon of salt in a large bowl. Cover with plastic wrap and microwave until the eggplant is tender and slightly translucent, about 10 minutes, stirring halfway through. Drain the eggplant in a colander and press with paper towels to remove excess moisture.

Whisk the crushed tomatoes, basil, olive oil, garlic, salt, and red pepper flakes together in a bowl and set aside.

Whisk the cottage cheese, cream, Parmesan, garlic, cornstarch, salt, and pepper together in a separate bowl until well combined.

Heat three tablespoons of olive oil in a large skillet over medium-high heat until shimmering. Add the zucchini and yellow squash and cook until they are browned and tender, about 8 to 10 minutes.

Stir in the garlic, thyme, and the eggplant. Add the spinach one handful at a time, stirring until each batch wilts and the liquid in the pan has evaporated.

Adjust the oven rack to the middle position and heat the oven to 375°F. Grease a 13 by 9-inch baking dish with the remaining teaspoon of olive oil.

Spread one cup of the tomato sauce over the bottom of the dish. Layer three noodles over the sauce, followed by a third of the vegetable mixture, a third of the olives, and a quarter of the mozzarella.

Drizzle half of the cream sauce over the cheese. Repeat the layering of noodles, tomato sauce, vegetables, olives, and mozzarella twice more.

Top with the final three noodles, the remaining tomato sauce, and the remaining mozzarella. Wrap the dish tightly with foil.

Bake until the sauce is bubbling and the noodles are tender, about 35 minutes. Remove the foil and continue to bake until the cheese is browned and spotty, about 15 minutes.

Let the lasagna rest for at least 15 minutes before slicing—this allows the layers to set so they don't slide apart. Sprinkle with fresh basil and serve warm.