Preheat your oven to 350°F (175°C). Grease a 9x13-inch baking dish with cooking spray or olive oil.
In a large mixing bowl, combine the shredded chicken, diced ham, and Swiss cheese. Mix well and set aside.
In another bowl, whisk together the heavy cream, chicken broth, Dijon mustard, garlic powder, onion powder, thyme, salt, and pepper until well combined.
Pour half of the cream mixture into the bottom of the prepared baking dish.
Layer the chicken and ham mixture evenly over the cream mixture in the baking dish.
Pour the remaining cream mixture over the top of the chicken and ham layer.
In a small bowl, combine the breadcrumbs with olive oil, mixing until the breadcrumbs are evenly coated. Sprinkle the breadcrumb mixture over the casserole.
Bake in the preheated oven for 30-35 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let it cool for 5-10 minutes before serving. Garnish with chopped parsley.