Preheat oven to 375°.
Heat a Dutch oven coated with cooking spray over medium-high heat. Add chard, squash, onion, bell pepper, carrot, and salt; saute 10 minutes or until tender.
Melt butter in a saucepan over medium heat. Add garlic; saute 30 seconds. Add flour, and cook 1 minute. Stir constantly with a whisk. Gradually add milk, stirring with a whisk until blended. Cook until thick (for about 4 minutes). Remove from heat; add 1/4 cup Parmesan. Stir until cheese melts.
Combine ricotta, cottage cheese, 1/2 cup Asiago, and oregano in a bowl.
Spread 2 tablespoons milk mixture in the bottom of an 8-inch square baking dish coated with cooking spray. Arrange 2 noodles over milk mixture; top with half of ricotta mixture, half of vegetable mixture, and 1/2 cup milk mixture. Repeat layers, ending with noodles. Spread remaining milk mixture over noodles. Sprinkle with 1/2 cup Asiago cheese and 2 tablespoons Parmesan. Bake at 375° for 45 minutes. Let stand 15 minutes.