Prepare the Chicken:
Flatten the chicken breasts to an even 1/4-inch thickness using a meat mallet. Season both sides of the chicken with kosher salt and black pepper.
Cook the Chicken:
Heat 3 tablespoons olive oil in a large skillet over medium heat. Once hot, add the chicken breasts without overcrowding the pan. Cook each side for 3-4 minutes until golden brown and fully cooked. Transfer the chicken to a baking pan and cover to keep warm.
Make the Sauce:
Using the same skillet (without washing), heat 2 tablespoons olive oil over medium heat. Add the sliced mushrooms and sauté for 1-2 minutes.
Add the garlic, Madeira wine, beef broth, cornstarch mixture, butter, and black pepper. Stir to combine and bring to a boil.
Reduce the heat to a simmer and cook for about 20 minutes, stirring occasionally, until the sauce is reduced by one-fourth and thickened to a dark brown consistency. Transfer the sauce to a container and keep warm.
Broil the Chicken:
Sprinkle the shredded mozzarella cheese evenly over the chicken breasts on the baking pan. Broil for 3-4 minutes, or until the cheese is melted and lightly golden. Watch carefully to prevent burning.
Serve:
Plate the chicken and generously drizzle with the Madeira mushroom sauce. Serve immediately with your favorite sides like mashed potatoes, rice, or pasta.