Snap off the tough ends of the asparagus, then bring 6 cups of water and 1 Tbsp salt to a boil in a medium pot and cook the asparagus uncovered for 2 to 3 minutes. The spears should turn bright green and stay crisp-tender, not soft or dull.

Set a large heavy oven-safe pan over medium/high heat and melt 2 Tbsp butter with 1 Tbsp oil, then cook the mushrooms for 5 minutes before adding the onion for 3 minutes. The mushrooms should soften and lose their raw look, and the onion should start to turn translucent.

Stir in the garlic, 1/4 tsp salt, 1/4 tsp black pepper, and 2 Tbsp parsley, then cook 2 minutes more and transfer everything to a plate before wiping the skillet clean with a wet paper towel. The mixture should smell deeply savory and garlicky, with no harsh raw garlic edge left.

Cut the chicken breasts in half lengthwise and pound each piece between plastic wrap until no thicker than 1/4 inch, then season with 1/2 tsp salt and 1/4 tsp black pepper. Even cutlets cook more quickly and brown more evenly, which helps keep them tender.

Return the same pan to medium/high heat with 1 Tbsp butter and 1 Tbsp oil, and once the butter stops foaming heavily, cook the chicken for 3 to 4 minutes per side until golden and cooked through. The surface should pick up a clear golden sear, and the center should reach 165°F for cooked chicken.

Move the chicken to the plate with the mushrooms, then pour 1 1/2 cups Madeira into the hot skillet and boil it vigorously for 5 minutes while scraping up the browned bits stuck to the pan. The liquid should smell sweet and savory, and this is where the sauce starts tasting like a restaurant pan sauce.

Add 1 1/2 cups beef broth and keep boiling for about 10 minutes, until only 2/3 cup remains or the liquid is reduced to about 1/4 of its starting volume. The sauce should look darker, more concentrated, and thick enough to lightly coat the back of a spoon.

Lower the heat to medium, stir in 1/2 cup cream, and simmer for 2 minutes, then season with salt and pepper to taste. The finished sauce should look silky and slightly thickened, not thin or watery.

Return the chicken to the pan and turn it in the sauce, then top with the mushrooms, asparagus, and 1 cup mozzarella cheese and broil for 3 to 4 minutes. The cheese should melt fully over the top, and the edges of the sauce will bubble around the chicken.

Take the skillet out of the oven and finish with fresh parsley. Serve while the sauce is glossy and the asparagus still has a little bite.