Preheat broiler. Bring a large pot of water to a boil and cook spaghetti one minute less than package directions.
Meanwhile, heat a small saucepan over medium heat. Melt butter then sprinkle in flour. Stir and cook for one minute. Add salt, pepper, thyme, nutmeg and Dijon mustard. Slowly whisk in milk and cook until sauce has thickened. Remove from heat and stir in 1 cup parmesan cheese.
In a large bowl, combine the chicken, pasta and sauce. Pour into a 13x9 pan, sprinkle with remaining parmesan cheese and all the cheddar cheese. Place under broiler until cheese melts.