Preheat oven to 375ºF. Place noodles in a 9-by-13-inch baking dish; cover with cold water. Move noodles around occasionally to prevent sticking.
Warm oil in a large pan over medium-low heat. Add onions, garlic and 1/4 tsp. salt. Sauté for 5 minutes. Stir in thyme and flour; cook, stirring, 3 minutes. Add milk in a steady stream, stirring; bring to a boil over medium-high heat. Reduce heat and simmer, stirring, until thickened, 3 minutes. Remove from heat; stir in 1/4 cup Parmesan, 3/4 tsp. salt and 1/4 tsp. pepper.
Drain noodles; place on a plate. Dry baking dish; mist with cooking spray. Spread 1 cup sauce on bottom. Line with 3 noodles; top with a third of squash. Cover with a third of remaining sauce. Sprinkle with 1/2 cup mozzarella. Repeat layering twice, starting with noodles and ending with mozzarella.
Cover dish tightly with foil. Bake until squash is tender, 35 minutes. Uncover. Sprinkle with remaining Parmesan and mozzarella; bake 15 minutes longer. Let cool for 10 minutes. Garnish with basil, if desired.