Preheat oven to 350°F.
Prepare the Mostaccioli and broccoli per package directions; drain and set aside in large bowl.
In a medium saucepan, melt butter. Stir in flour until well blended, about 2 minutes. Add milk all at once. Cook until thickened, stirring constantly, about 10 minutes. Stir in Parmesan cheese, salt, garlic salt, thyme leaves, and nutmeg. Remove from heat.
Add the chicken strips to the Mostaccioli and broccoli. Pour alfredo sauce over the top and combine until well-coated.
Spoon mixture into a 12×9-inch baking dish. Sprinkle Swiss cheese on top.
Bake at 350°F for 10-15 minutes or until thoroughly heated.
Sprinkle with fresh grated Parmesan cheese and serve.