Cook milk in a small, heavy saucepan over medium-high heat to 180° or until tiny bubbles form around edge (do not boil). Remove from heat; keep warm.
Lightly spoon flour into a dry measuring cup; level with a knife. Melt butter in a medium nonstick saucepan over medium heat. Add shallots; cook 2 minutes or until tender. Reduce heat; add flour to pan, and cook 5 minutes or until smooth and golden, stirring constantly. Remove from heat; add about 2 tablespoons warm milk to flour mixture, stirring constantly with a whisk. Gradually add remaining warm milk, about 1/2 cup at a time, until mixture is smooth, stirring constantly with a whisk. Stir in 1/2 teaspoon salt and 1/4 teaspoon black pepper. Bring to a boil; reduce heat, and cook until smooth and thickened. Remove from heat. Cover surface of milk mixture with plastic wrap; set aside.
Preheat oven to 425°.
Place squash in a large bowl. Add vinegar; toss to coat. Add 1 tablespoon oil; toss to coat. Arrange squash in a single layer on a jelly-roll pan coated with cooking spray. Sprinkle with 1/2 teaspoon salt, remaining 1/4 teaspoon black pepper, and thyme. Bake at 425° for 30 minutes, stirring after 15 minutes.
Combine remaining 1 teaspoon oil, red pepper, and garlic in a Dutch oven over medium heat; cook 2 minutes, stirring constantly. Add spinach, 1 bag at a time; cook until wilted, stirring frequently. Add remaining 1/4 teaspoon salt; cook until liquid evaporates, stirring frequently.
Reduce oven temperature to 350°.
Spoon 1/3 cup milk mixture in bottom of a 13 x 9-inch baking pan coated with cooking spray. Arrange 3 noodles over milk mixture; top with spinach mixture, 2/3 cup milk mixture, 1/2 cup Asiago, and 1/4 cup Parmigiano-Reggiano. Arrange 3 noodles over cheese; top with squash mixture, 2/3 cup milk mixture, remaining 1/2 cup Asiago, and 1/4 cup Parmigiano-Reggiano. Arrange remaining 3 noodles on top of cheese; spread remaining 1/2 cup milk mixture over noodles. Sprinkle with the remaining 1/2 cup Parmigiano-Reggiano. Bake at 350° for 30 minutes or until bubbly. Let stand 15 minutes before serving.