Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking dish with olive oil.
In a large mixing bowl, combine the cooked shredded chicken, broccoli, cauliflower, mushrooms, and half of the Swiss cheese.
In another bowl, whisk together the heavy cream, chicken broth, eggs, minced garlic, onion powder, thyme, salt, and pepper until well blended.
Pour the cream mixture over the chicken and vegetables, stirring to combine all the ingredients evenly.
Transfer the mixture into the greased baking dish, spreading it out evenly.
Sprinkle the remaining Swiss cheese over the top of the casserole.
Bake in the preheated oven for 30-35 minutes or until the cheese is bubbly and golden brown.
Remove from the oven and let it cool for a few minutes before serving.