Preheat the oven to 375°F and spray a 13x9-inch baking dish with nonstick spray.
Heat a large Dutch oven or large skillet over medium-high heat and coat with cooking spray or a little oil. Sauté the chopped onion 4 minutes until lightly browned, then add the minced garlic and cook 1 minute more. Transfer the mixture to a large bowl.
Heat 1 teaspoon of the olive oil in the pan over medium-high heat. Add the chopped zucchini and yellow squash and sauté about 4 minutes until tender and beginning to brown. Add to the onion and garlic in the bowl.
Heat the remaining 1 teaspoon oil in the pan over medium-high heat. Add the thinly sliced carrots and sauté 4 minutes until tender. Add the chopped broccoli and sauté an additional 4 minutes until crisp-tender. Add these vegetables to the bowl with the others and sprinkle with 1/2 teaspoon salt; toss to combine.
Make the white sauce: whisk the flour into a medium saucepan, then gradually whisk in the milk until smooth. Bring to a boil over medium heat, stirring constantly, and cook about 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 thawed, well-drained spinach.
Combine the cottage cheese and 1½ cups of the shredded mozzarella in a bowl and stir until blended.
Assemble the lasagna: spread 1/2 cup of the sauce in the bottom of the prepared baking dish. Arrange 3 noodles over the sauce. Top with half the cottage cheese–mozzarella mixture, half the vegetable mixture, and about 1 cup of the sauce. Repeat the layer with 3 more noodles, the remaining cottage cheese mixture, remaining vegetables, and about 1 cup sauce. Top with the last 3 noodles, spread the remaining sauce over the noodles, then sprinkle with the remaining 1/2 cup Parmesan and the remaining 1/2 cup shredded 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 the lasagna stand 10 minutes before slicing and serving.