Slice chicken breasts in half lengthwise. Use a meat mallet to pound each piece of chicken to ¼" thickness between two pieces of plastic wrap. Season chicken on both sides with ½ teaspoon salt and ¼ teaspoon pepper. Set aside.
In a large skillet, melt 2 tablespoons of butter and 1 tablespoon of oil over medium-high heat. Add the sliced mushrooms and season with ¼ teaspoon salt and ¼ teaspoon pepper. Cook for 5 minutes, stirring occasionally, until softened. Add the garlic for the last minute, then transfer cooked mushrooms to a plate.
In the same skillet, melt 1 additional tablespoon each of butter and oil. Saute the chicken breasts for 3-4 minutes on each side until nicely browned and cooked through. Transfer to the plate with the mushrooms.
In the same skillet, add the Madeira wine and bring to a boil. Boil vigorously for 5 minutes until reduced by half, using a wooden spoon to scrape up any browned bits left over from cooking the chicken from the bottom of the pan.
Once the Madeira has reduced by half, add the beef broth and bring to a boil. Boil for about 10 minutes until only about 1 cup of liquid remains (about ¼ of the original volume of liquid). Reduce heat to medium-low and stir in the cream. Let the sauce simmer for 2 minutes until slightly thickened. Taste and adjust salt and pepper as needed.
While the Madiera sauce is reducing, blanch the asparagus by adding it to a large pot of salted, boiling water and cooking for 2-3 minutes, just until tender-crisp and bright green. Drain immediately and set aside.
Return the cooked chicken and mushrooms to the pan, tossing to coat in the Madeira sauce. Arrange the blanched asparagus around the chicken and sprinkle with mozzarella cheese. Place the skillet in the oven under the broiler for 3-4 minutes, just until the cheese is melted, then serve with mashed potatoes.