Pound chicken breasts until they’re an even ¼-inch thickness throughout. Generously season both sides of each chicken breast with salt and pepper.
Heat 1 Tbsp. olive oil in a large cast-iron skillet. Place chicken breasts in pan and cook until browned on both sides and mostly cooked through. Transfer to a baking pan and cover with foil to keep warm.
Add (to the same skillet) another Tbsp. of olive oil. Heat and add sliced mushrooms. Saute until they begin to soften.
Add the rest of the sauce ingredients. . Bring sauce to a boil, then immediately reduce heat to a simmer. Simmer for 5 minutes, then add the chicken breasts to the pan and allow to finish cooking while sauce is simmering. (This should take about 15 minutes. The sauce should be thickened and reduced by about one-fourth of its original volume. If chicken cooks completely before sauce is fully reduced, remove from pan and cover with foil to keep warm.)
Bring medium saucepan with water to boil. Toss asparagus into water and boil for 3-5 minutes. Drop asparagus in a bowl of ice water to halt the cooking. It should be slightly tender.
If using mozzarella cheese, remove chicken from pan and place back on the baking pan. Place asparagus on top and then sprinkle cheese on top and broil for 3-4 minutes until cheese is melted and bubbly.
Serve chicken immediately, generously drizzled with sauce. This goes great with white rice or potatoes and roasted veggies!