Flatten chicken breasts to an even ¼-inch thickness throughout. Lightly sprinkle both sides of each chicken breast with kosher salt and black pepper.
In a large skillet, heat 3 TB olive oil over medium heat until hot. Place chicken breasts in skillet, without overcrowding. Cook 3-4 minutes per side, until nicely browned on both sides and no longer pink in center. Transfer cooked chicken to a baking pan and cover to keep warm.
In the same unwashed skillet, add 2 TB olive oil over medium heat until oil is hot. Add mushrooms and stir 1-2 min. Add the remaining sauce ingredients. Stir and bring sauce to a boil.
Immediately reduce heat to a simmer. Simmer 20 minutes or until sauce is reduced by one-fourth of its original volume. Finished sauce will be dark brown and thickened. Transfer sauce to a container and keep warm.
Sprinkle mozzarella cheese evenly on top of chicken breasts on baking pan. Broil 3-4 minutes or just until cheese is golden and melted; watch carefully to prevent burning.
Enjoy chicken immediately, generously drizzled with Madeira sauce. Extra sauce is delicious over mashed potatoes, angel hair pasta, or fluffy rice.
