01 - Preheat the 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, black pepper, and dried thyme.
03 - In a large skillet over medium heat, melt the butter. Add the chopped onion and minced garlic, sautéing 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 over the cooked mushrooms and stir continuously for 1 minute to form a light roux and cook off the raw flour taste.
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 the prepared baking dish in a single layer. Pour the mushroom-cheese sauce evenly over each piece of chicken.
09 - Sprinkle the remaining mozzarella cheese evenly over the top of the assembled casserole.
10 - Bake uncovered for 30 to 35 minutes, 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 to allow the sauce to set. Garnish with chopped fresh parsley before serving.