Preheat the oven to broil and place a rack in the highest position. Arrange the eggplant in a single layer on a baking sheet and spray or brush lightly with olive oil. Toss to coat, then broil, tossing once, until soft and brown, about 15 minutes.
When the eggplant is done, preheat the oven to bake at 375ºF and move the oven rack to the middle position. Spray a 9x13 baking dish with cooking spray or olive oil and pour in about ½ cup of marinara. Spread the marinara to lightly coat the bottom of the pan and set aside.
Meanwhile, bring a large pot of water to a boil and cook pasta according to package directions. When it’s done, drain into a colander in the sink and place under cold running water until cool enough to handle. Lay each noodle out on a clean kitchen towel or paper towel and pat dry.
While the pasta is cooking, heat 1 tablespoon olive oil in a large pan over medium heat. Add the onion and cook until translucent, about 7 minutes. Add the zucchini, yellow squash, and tomato and cook, stirring, two minutes more. Add the spinach and garlic and continue to cook and stir until the spinach is wilted and the garlic is fragrant, about 1 minute more. Remove from the heat. If a lot of liquid has accumulate, drain a little off.
In a large bowl, combine ricotta, egg, eggplant, onion mixture, and ½ cup Parmesan. Season to taste with salt and pepper. Spoon about 2 tablespoons of the filling onto each lasagna noodle, leaving about ½ an inch of space at the end. Roll the lasagna up and place seam side down in the prepared baking dish. Cover with the remaining marinara sauce and sprinkle with remaining cheeses. Cover the pan with aluminum foil and bake 25-30 minutes or until bubbly and heated through.