Preheat the oven to 350°F (175°C).
In a large mixing bowl, combine the shredded chicken, diced ham, and ½ cup of Swiss cheese. Mix well.
In another bowl, whisk together the sour cream, chicken broth, Dijon mustard, garlic powder, black pepper, salt, and dried thyme until smooth and well combined.
Pour the sour cream mixture over the chicken and ham mixture. Stir until all the ingredients are evenly coated.
Transfer the mixture into a greased 9x13 inch baking dish, spreading it out evenly.
Sprinkle the remaining Swiss cheese over the top of the casserole.
Finish by adding the breadcrumb topping evenly over the cheese to create a crunchy topping.
Bake in the preheated oven for 30 minutes, or until the casserole is hot and bubbly and the breadcrumbs are golden brown.
Once done, remove from the oven and let it cool for a few minutes. Garnish with chopped green onions if desired.