On a hot griddle or skillet, cook the chicken over medium heat until cooked through. Chop chicken into 1-inch pieces. Set aside.
Over medium heat, bring a pot of water to a boil. When it boils, sprinkle in 1/2 teaspoon salt and add the spaghetti and carrots. Drain completely when spaghetti is al dente. Transfer the chicken pieces, spaghetti, and carrots to a large casserole pan. Stir in the bechamel sauce (below), remaining 1/2 teaspoon salt, celery, chicken broth, parmesan cheese, and black pepper.
If desired, sprinkle with additional parmesan cheese.
BECHAMEL SAUCE: In a medium heavy saucepan, melt the butter over low heat. Add the flour. Whisk continuously until the roux is thickened. Add the milk gradually, whisking continuously. Sprinkle the in the nutmeg and salt. When the bechamel sauce becomes thick, remove it from the heat. NOTE: If the sauce cools and you wish to thin it out, just add a little more milk.
Cover loosely with foil and bake for 25 - 30 minutes or until heated through.