Preheat the oven to 375°F and spray a 13×9-inch baking dish with nonstick spray.
Heat a large Dutch oven or skillet over medium-high and coat with cooking spray; sauté the chopped onion 4 minutes until lightly browned, then add the minced garlic and sauté 1 minute. Transfer to a large bowl.
Heat 1 teaspoon of the olive oil over medium-high; sauté the chopped zucchini and yellow squash about 4 minutes until tender and lightly browned, then add to the onion mixture.
Heat the remaining 1 teaspoon oil over medium-high; sauté the thinly sliced carrot about 4 minutes until tender, add the chopped broccoli and cook 4 minutes until crisp-tender. Add these vegetables to the bowl with the others, sprinkle with 1/2 teaspoon salt and toss to combine.
In a medium saucepan, place the flour and gradually whisk in the milk until smooth. Bring to a boil over medium heat, stirring constantly, and cook 2 minutes until thickened. Remove from heat and stir in 1/2 cup Parmesan, 1/2 teaspoon salt, 1/4 teaspoon black pepper, and a dash of nutmeg until smooth. Stir in the squeezed dry spinach.
In a bowl, combine the cottage cheese with 1½ cups of the shredded mozzarella and stir well.
Spread 1/2 cup of the sauce in the bottom of the prepared baking dish. Arrange 3 lasagna noodles over the sauce. Top with half the cottage cheese mixture, half the vegetable mixture, and about 1 cup of the sauce. Repeat the layers with 3 more noodles, the remaining cottage cheese mixture, the remaining vegetables, and about 1 cup sauce. Finish with the last 3 noodles, spread the remaining sauce on top, then sprinkle with 1/2 cup Parmesan and the remaining 1/2 cup mozzarella.
Cover the dish and bake 20 minutes. Uncover and bake an additional 20 minutes until the cheese is bubbly and beginning to brown. Let stand 10 minutes before serving.