In a large ovenproof skillet, heat 1 tablespoon of olive oil and 2 tablespoons of butter over medium-high heat.
Season the chicken cutlets with Kosher salt and black pepper.
Add the seasoned chicken to the skillet, cooking for 4 to 5 minutes per side or until cooked through. Once done, transfer the cooked chicken to a plate and keep it aside.
Return the same skillet to medium-high heat. Melt the remaining 1 tablespoon butter and 1 tablespoon olive oil, and add sliced mushrooms. Season with a pinch of salt and cook for 2 to 3 minutes until soft, allowing them to develop a golden-brown color on one side.
Stir the mushrooms to ensure even cooking, then add the finely diced onion to the skillet.
Sauté the mushrooms and onions, stirring occasionally, until the mushrooms are tender, and the onions are translucent, which should take approximately 4 to 5 minutes.
About a minute before finishing, add the minced garlic to the skillet, stirring until the garlic is fragrant. Transfer the sautéed mixture to a plate, covering it to keep warm.
Preheat your oven's broiler to high.
Return the skillet to medium-high heat. Add the Madeira wine, stirring to deglaze the pan. Bring it to a boil and continue boiling for 3 minutes.
Whisk in the low-sodium beef broth, bring to a boil, and cook for an additional 7 to 10 minutes or until the sauce is reduced.
Lower the heat to medium, whisk in the heavy cream, and let it simmer for 1 to 2 minutes until the sauce thickens.
Taste the sauce for salt and pepper, adjusting as needed.
While the sauce simmers, return the cooked chicken, sautéed mushrooms, and blanched asparagus to the skillet.
Sprinkle the tops with shredded mozzarella cheese.
Place the skillet under the preheated broiler for 3 minutes or until the cheese is melted and golden.
Remove from the oven, garnish with additional fresh parsley, and serve immediately.