1. Preheat your oven to 375°F (190°C). Lightly grease a baking dish large enough to hold the chicken breasts snugly.
2. Season the chicken breasts with salt and pepper. Using a sharp knife, make a pocket inside each chicken breast by slicing horizontally but not all the way through.
3. In a skillet over medium heat, add the olive oil. Sauté the chopped mushrooms until they are golden and tender. Remove and set aside.
4. In the same skillet, melt the butter and add the minced garlic. Cook until fragrant, about 1 minute. Stir in the cooked mushrooms, ensuring they’re well coated with the garlicky butter.
5. Stuff each chicken breast with the mushroom mixture and secure with toothpicks if necessary.
6. Place the stuffed chicken breasts in the prepared baking dish. Bake for 20-25 minutes or until the chicken is cooked through and juices run clear.
7. While the chicken is baking, prepare the creamy Parmesan sauce. In a saucepan over medium heat, combine the heavy cream, Parmesan cheese, thyme, and chicken broth. Stir until the sauce is smooth and slightly thickened, about 5 minutes.
8. Once the chicken is done, pour the creamy Parmesan sauce over the baked chicken. Sprinkle with fresh parsley before serving.

This recipe has a touch of elegance but is straightforward enough to make on a busy evening. Remember, cooking is as much about enjoying the process as it is about the end result, so feel free to sip a glass of your favorite wine while the chicken bakes.