Preheat the oven to 375°F. Spray a 13x9-inch baking dish with nonstick spray.
PREPARE THE VEGGIES:
Heat a large Dutch oven over medium-high heat. Coat the pan with cooking spray. Add the onion to the pan; sauté 4 minutes or until lightly browned. Add the garlic; sauté 1 minute. Spoon the onion mixture into a large bowl.
Heat 1 teaspoon of the oil in a pan over medium-high heat. Add the zucchini and yellow squash; sauté 4 minutes or until tender and just beginning to brown. Add to the onion mixture.
Heat the remaining 1 teaspoon of oil in a pan over medium-high heat. Add the sliced carrot; sauté 4 minutes or until tender. Add the chopped broccoli; sauté 4 minutes or until crisp-tender. Add to the onion mixture. Sprinkle with ½ teaspoon salt; toss well to combine.
PREPARE THE SAUCE:
Place the flour in a medium saucepan. Gradually add the milk, stirring with a whisk until blended. Bring to a boil over medium heat; cook 2 minutes or until thick, stirring constantly. Remove from heat. Add the Parmesan, salt, pepper, and nutmeg; stir until smooth. Stir in the spinach.
ASSEMBLE THE LASAGNA:
Combine the cottage cheese and 1½ cups mozzarella; stir well. Spread ½ cup of the sauce mixture in the bottom of the prepared baking dish. Arrange 3 noodles over the sauce mixture in the dish; top with half of the cottage cheese mixture, then half of vegetable mixture, and about 1 cup of the sauce mixture. Repeat layers, ending with noodles. Spread the remaining sauce mixture on top of the noodles; sprinkle with ½ cup Parmesan and remaining ½ cup mozzarella.
Cover and bake for 20 minutes. Uncover and bake an additional 20 minutes or until the cheese is bubbly and beginning to brown. Let stand 10 minutes before serving.