Bring stock to a boil in a large saucepan over medium-high heat. Add chicken; reduce heat, and simmer 18 to 20 minutes or until chicken is done. Remove chicken from pan; reserve stock. When cool enough to handle, shred with 2 forks.
Heat a large skillet over medium-high heat. Add oil; swirl to coat. Add onion, garlic, 1/2 teaspoon salt, thyme, pepper, and mushrooms; sauté 8 minutes. Stir in spinach, 1 package at a time, cooking 2 minutes after each addition or until spinach wilts before adding more. Remove pan from heat; stir in chicken.
Preheat oven to 375°.
Coat 2 (8-inch) square baking dishes with cooking spray. Combine milk, flour, and nutmeg, stirring with a whisk until smooth. Bring reserved stock to a boil over medium-high heat. Gradually add milk mixture to stock mixture, stirring constantly with a whisk. Stir in remaining 1 teaspoon salt. Cook 5 minutes or until thickened, stirring occasionally. Remove pan from heat; stir in Parmigiano-Reggiano. Pour 1/2 cup sauce into bottom of each baking dish. Top each with 2 noodles, 1 1/2 cups chicken mixture, 3/4 cup sauce, and 1/4 cup mozzarella, making sure noodles are covered with sauce. Repeat layers once with noodles, chicken mixture, and sauce. Top dishes evenly with remaining 4 noodles, remaining 1/2 cup sauce, and remaining 3/4 cup mozzarella.
Cover dishes tightly with foil coated with cooking spray. Bake at 375° for 25 minutes. Uncover and bake at 375° for 10 to 15 minutes or until browned and bubbly, or follow freezing instructions. Let stand 5 minutes before serving.
How-to Freeze: Cool pan completely. Cover with foil; freeze up to 2 months. Thaw: In metal pan: Thaw overnight in refrigerator. In glass or ceramic: Uncover and microwave at HIGH 15 minutes or until thawed. Reheat: Cover; bake at 375° for 25 minutes. Uncover and bake 5 minutes.