Preheat the oven to 350°F (175°C). Grease a 9x13-inch baking dish with cooking spray or butter.
In a large bowl, combine the shredded chicken, diced ham, and half of the Swiss cheese. Mix well and spread the mixture evenly in the prepared baking dish.
In a separate bowl, whisk together the heavy cream, chicken broth, Dijon mustard, garlic powder, onion powder, black pepper, and salt until well combined.
Pour the cream mixture evenly over the chicken and ham layer in the baking dish.
In another bowl, mix the breadcrumbs with the remaining Swiss cheese. Sprinkle this mixture evenly over the top of the casserole.
Bake in the preheated oven for 30-35 minutes, or until the top is golden brown and the casserole is bubbly.
Remove from the oven and let it cool for about 5 minutes before serving. Garnish with fresh parsley if desired.