Preheat the oven to 375°F (190°C) and lightly spray a 13×9-inch baking dish with nonstick spray.
Heat a large skillet or Dutch oven over medium-high heat and coat with cooking spray; sauté the chopped onion 4 minutes until lightly browned, then add the minced garlic and cook 1 more minute. Transfer to a large bowl.
Heat 1 teaspoon of the olive oil 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 bowl with the onion and garlic.
Heat the remaining 1 teaspoon oil over medium-high heat; sauté the thinly sliced carrot 4 minutes until tender, then add the chopped broccoli and cook about 4 minutes until crisp-tender. Add to the vegetable mixture, sprinkle with 1/2 teaspoon salt, and toss to combine.
In a medium saucepan, place the flour 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 pepper, and a dash of nutmeg until smooth.
Stir the well-drained spinach into the sauce, then mix the cottage cheese with 1½ cups of the shredded mozzarella until combined.
Spoon 1/2 cup of the sauce into the bottom of the prepared baking dish. Arrange 3 lasagna noodles over the sauce. Top with half of the cottage cheese–mozzarella mixture, half of the vegetable mixture, and about 1 cup of the sauce.
Repeat the layers: 3 noodles, remaining cottage cheese mixture, remaining vegetables, and about 1 cup sauce. Top with the last 3 noodles, spread the remaining sauce over them, then sprinkle with 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 cheese is bubbly and lightly browned. Let the lasagna stand 10 minutes before serving.