Place chicken, 1 teaspoon salt, and water to cover in a Dutch oven. Bring to a boil; cover and simmer 45 minutes. Remove chicken from broth; cool, reserving 2 1/2 cups broth. Bone chicken, and cut meat into bite-size pieces. Set aside.
Place tomatoes in a heavy saucepan. Bring to a boil; reduce heat, and simmer 15 minutes. Set aside.
Melt 3 tablespoons butter in a heavy saucepan over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually stir in cream and 1 cup reserved broth; cook over medium heat, stirring constantly, until thickened and bubbly. Stir in nutmeg, pepper, and remaining salt. Set aside.
Combine beef and sausage in a medium skillet; cook over medium heat until meat is browned, stirring to crumble. Drain and set aside.
Sauté mushrooms, onion, celery, green pepper, and garlic in remaining butter in a Dutch oven 5 minutes or until vegetables are crisp-tender. Add tomatoes, white sauce, meat, bay leaf, and red pepper flakes to Dutch oven; stir well. Simmer over low heat 15 minutes. Discard bay leaf. Cook spaghetti according to package directions; drain and set aside.
Spoon one-third of meat sauce into a lightly greased 5- quart casserole. Top with half of spaghetti, half of chicken, one-third of meat sauce, and half of Cheddar cheese. Repeat layers, beginning with spaghetti and ending with Cheddar cheese. Pour remaining reserved broth over layers. Bake, covered, at 350° for 15 minutes. Uncover and bake an additional 15 minutes. Serve with Parmesan cheese.