Season chicken cutlets with Italian seasoning, paprika, onion powder, salt, and pepper.
Heat 1 tablespoon olive oil with 1 tablespoon butter in a large Dutch oven skillet set over medium-high heat.
Add chicken breast to the skillet, and cook for 4-5 minutes per side or until cooked through and internal temperature registers at 165˚F (74℃) on an instant-read meat thermometer. Transfer the chicken to a plate and keep it covered.
Return the skillet to medium-high heat and add the remaining butter and olive oil.
Add mushrooms to the skillet, season with salt, and saute for 4-5 minutes or until soft.
Stir in minced garlic and thyme, and continue cooking and stirring for 30 seconds. Transfer mushrooms to a plate, set aside, and keep covered.
Return skillet over medium-high heat.
Deglaze the skillet with the wine, and stir all the browned bits from the bottom of the pan. Bring to a boil, and simmer for 5 minutes until reduced to about half the quantity.
Whisk in chicken broth and bring to a boil. Continue to cook for 5-8 more minutes or until sauce reduces.
Reduce heat to medium and whisk in the heavy cream. Let simmer for 1-2 minutes or until sauce thickens. Taste for salt and pepper, and adjust accordingly.
In the meantime, set your broiler to high.
Return chicken, mushrooms, and asparagus to the skillet. Sprinkle tops with shredded cheese.
Broil for 3-4 minutes or until cheese melts.
Remove from oven, garnish with parsley, and serve with rice or mashed potatoes.