Copycat Cheesecake Factory Madeira Chicken

Kosher salt
1 bunch asparagus, woody ends removed
2 large chicken breasts (about 1 lb. total), each cut into 2 cutlets and pounded ½” thick
1/4 tsp. freshly ground black pepper
4 tbsp. extra-virgin olive oil
1/2 medium yellow onion, chopped (about 1 cup)
1 lb. white button or cremini mushrooms, sliced
2 garlic cloves, minced
2 tbsp. all-purpose flour
1/2 c. Madeira wine
1 c. low-sodium beef broth
1 tbsp. lemon juice
2 tbsp. chopped fresh parsley, plus more for garnish
1 c. shredded mozzarella

Bring a large skillet filled with water to a boil and add 2 tablespoons of salt. Add asparagus and cook just until bright green and slightly tender, 3 to 4 minutes. Drain and transfer asparagus to a plate. Return skillet to the stove. (Make sure skillet is dry before proceeding to next step.)
Season chicken cutlets on both sides with ½ teaspoon salt and ¼ teaspoon black pepper. Add 2 tablespoons oil to the skillet and heat over medium high. Once the oil shimmers, sear cutlets until golden brown and cooked through, about 2 to 3 minutes per side. Transfer chicken to a plate and set aside.
In the same skillet, heat 1 tablespoon oil until it shimmers. Add onions and season with ½ teaspoon salt. Cook, stirring frequently, until softened slightly, about 4 to 5 minutes. Add mushrooms and season again with ½ teaspoon salt. Cook, stirring occasionally, until mushrooms soften and release much of their moisture, about 5 to 7 minutes. Add minced garlic and cook just until garlic is fragrant, about 1 minute more.
Add 1 tablespoon of oil to the skillet, then add the flour. Stir well to evenly coat mushroom mixture and to cook raw flour, about 2 minutes. Add Madeira and scrape/stir well, dislodging the brown bits from the skillet, about 30 seconds. Add beef broth and lower the heat to medium, stirring continuously to thicken the sauce. Add lemon juice and parsley, and season to taste with salt and pepper.
Preheat broiler to high. Return chicken cutlets and any accumulated juices to the pan with the sauce and turn to coat. Add asparagus to the skillet and sprinkle evenly with mozzarella. Broil until mozzarella is melted and golden in spots, about 2 minutes. Remove from the oven and garnish with parsley.