In a large skillet, heat the olive oil over medium heat. Add the chopped onion and sauté until translucent, about 3 minutes.
Stir in the minced garlic and sliced mushrooms, cooking until the mushrooms are softened, approximately 5 minutes.
In a bowl, combine the heavy cream, dried thyme, salt, and pepper. Mix well and set aside.
In a greased baking dish, arrange the chicken breasts in a single layer. Pour the cream mixture over the chicken, making sure to coat each piece evenly.
Spread the sautéed mushroom and onion mixture over the chicken.
Evenly sprinkle the shredded Swiss cheese on top of the mushrooms and onions.
Finally, sprinkle the breadcrumbs over the entire dish for a crunchy topping.
Bake in the preheated oven for 30-35 minutes, or until the chicken is cooked through and the breadcrumbs are golden brown.
Remove from the oven and let it cool for a few minutes before garnishing with freshly chopped parsley.