Preheat oven to 375°.
Heat a large Dutch oven over medium-high heat. Coat pan with cooking spray. Add onion to pan; sauté 4 minutes or until lightly browned. Add garlic; sauté 1 minute. Spoon onion mixture into a large bowl.
Heat 1 teaspoon oil in pan over medium-high heat. Add zucchini and yellow squash; sauté 4 minutes or until tender and just beginning to brown. Add to onion mixture.
Heat remaining 1 teaspoon oil in pan over medium-high heat. Add sliced carrot; sauté 4 minutes or until tender. Add chopped broccoli; sauté 4 minutes or until crisp-tender. Add to onion mixture. Sprinkle with 1/2 teaspoon salt; toss well to combine.
Place flour in a medium saucepan. Gradually add 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 1/2 cup Parmesan, remaining 1/2 teaspoon salt, pepper, and nutmeg; stir until smooth. Stir in spinach.
Combine cottage cheese and 1 1/2 cups mozzarella; stir well. Spread 1/2 cup spinach mixture in bottom of a 13 x 9-inch baking dish coated with cooking spray. Arrange 4 noodles over spinach mixture in dish; top with half of cottage cheese mixture (about 1 1/2 cups), half of vegetable mixture (about 2 1/2 cups), and about 1 cup spinach mixture. Repeat layers, ending with noodles. Spread remaining spinach mixture over noodles; sprinkle with remaining 1/2 cup Parmesan and remaining 1/2 cup mozzarella.
Cover and bake at 375° for 20 minutes. Uncover and bake an additional 20 minutes or until cheese is bubbly and beginning to brown. Let stand 10 minutes before serving.