01 - Preheat oven to 400°F. Lightly grease a 9x13-inch baking dish with butter or cooking spray.
02 - Pat chicken breasts dry and season both sides evenly with salt, pepper, and dried thyme.
03 - In a large skillet over medium heat, melt the butter. Add the chopped onion and minced garlic; sauté for 2 to 3 minutes until softened and fragrant.
04 - Add the sliced mushrooms to the skillet and cook for 5 to 6 minutes, stirring occasionally, until golden brown and all released liquid has evaporated.
05 - Sprinkle the flour evenly over the mushroom mixture. Stir continuously and cook for 1 minute to form a light roux.
06 - Gradually pour in the chicken broth and heavy cream, stirring constantly to prevent lumps. Bring to a gentle simmer and cook for 2 to 3 minutes until the sauce slightly thickens.
07 - Remove the skillet from heat. Stir in half of the mozzarella and all of the Parmesan cheese until fully melted and smooth.
08 - Arrange the seasoned chicken breasts in a single layer in the prepared baking dish. Pour the mushroom-cheese sauce evenly over the chicken, ensuring full coverage.
09 - Sprinkle the remaining mozzarella cheese evenly across the top of the casserole.
10 - Bake uncovered for 30 to 35 minutes, or until the chicken reaches an internal temperature of 165°F and the top is bubbly and golden brown.
11 - Remove from the oven and let rest for 5 minutes. Garnish with freshly chopped parsley before serving.