Preheat your oven to 375 degrees F (190 degrees C). Lightly grease a 9x13 inch baking dish.

In a large skillet over medium heat, melt the butter.

Add the onion and mushrooms. Cook for 5 to 7 minutes, stirring occasionally, until the mushrooms are tender and their moisture has cooked away.

Add the garlic and cook for another 30 seconds.

Stir in the cream of mushroom soup, sour cream, chicken broth, Dijon mustard, thyme, salt, and black pepper. Cook for 2 to 3 minutes until smooth and heated through.

Fold the cooked shredded chicken into the sauce until everything is evenly coated.

Spread the chicken mixture into the prepared baking dish. Arrange Swiss cheese slices over the top, then sprinkle with mozzarella.

Cover the dish loosely with foil and bake for 20 minutes.

Remove the foil and continue baking for 10 to 15 minutes, until the cheese is melted, bubbly, and lightly golden around the edges.

Let the casserole rest for 5 minutes before serving. If desired, sprinkle with fresh parsley.