Cook noodles as directed on package, drain. Heat oven to 375F. Grease rectangular pan, 11 x 7 1 1/2 inches. Meanwhile heat butter in 2-quart saucepan over medium heat until melted. Stir in flour and pepper. Cook over medium heat, stirring constantly, until smooth and bubbly; remove from heat. Stir in milk and broth. Heat to boiling, stirring constantly. Boil and stir 1 minute.
Mix spinach, noodles, half of the sauce, 1/4 cup cheese and 1/4 cup nutmeg. Spoon mixture into pan. Place chicken on top. Pour remaining sauce over chicken and spinach mixture. Sprinkle with 1/4 cup cheese and additional nutmeg. Cover with aluminum foil and bake 30 minutes.
Uncover and bake about 15 minutes longer or until light brown on top and juices of chicken run clear.