Slice each chicken breast lengthwise into two equal pieces. Place each piece between two sheets of heavy-duty plastic wrap and pound to a 1/4-inch thickness using a meat mallet. Repeat with all chicken pieces.

In a large skillet, melt 2 tablespoons of butter over medium heat. Add the sliced mushrooms and sauté for 3-5 minutes until tender. Transfer the cooked mushrooms to a plate or bowl and set aside.

Place the flour on a flat plate and coat each piece of chicken evenly. In the same skillet, melt 2 tablespoons of the remaining butter over medium-high heat. Add two pieces of chicken and cook until browned on one side. Flip and brown the other side, then transfer to a plate. Repeat with the remaining chicken, adding more butter as needed.

Once all chicken pieces are browned, stir the chicken broth and Marsala wine into the skillet drippings. Scrape the bottom to deglaze the pan, and season with salt and pepper. Bring to a boil, then reduce heat and simmer for 5 minutes, allowing the sauce to thicken slightly.

In a small bowl, combine the cornstarch and water to create a slurry. Whisk the slurry into the skillet sauce and continue to simmer for an additional 1-2 minutes until thickened. Remove the skillet from heat.

Arrange the chicken breasts in a lightly greased 9x13-inch baking dish, slightly overlapping if necessary. Distribute the sautéed mushrooms over the chicken.

Pour the prepared sauce evenly over the chicken, then sprinkle with mozzarella and Parmesan cheese. Top with sliced green onions.

Bake at 450°F (232°C) for 15-20 minutes, or until the cheese is melted and golden. Serve hot.