fresh thyme or parsley and cracked black pepper for topping(optional)
Preheat oven to bake at 400 degrees OR broil on low. In a large oven-safe skillet (see note) over medium-high heat, melt butter. Add onions and 3 tablespoons beef broth and saute onions for 3-4 minutes until translucent. Continue to cook, stirring occasionally so the onions don't burn, for about 15 minutes longer until browned and very tender. Use tongs or a fork to transfer to a bowl and cover to keep warm.
While onions are cooking, prepare the chicken by drizzling with oil, then seasoning with salt and pepper (to taste) and Italian herbs. Once onions have finished cooking and are removed from the pan, cook chicken for 4-5 minutes on each side (don't clean out the pan between the onions and chicken) until browned on both sides. (Chicken may not be fully cooked through yet, that is okay).
Transfer chicken to a plate and cover to keep warm and return the onions to the pan. Sprinkle flour over the onions and stir for 1 minute over medium-high heat. Add beef broth and continue to cook, stirring throughout, until mixture comes to a boil. Season with salt and pepper to taste. Return chicken to pan and spoon some of the sauce over each piece of chicken.
Top chicken with one slice of provolone each, then one slice of swiss, then 1/4 of the parmesan cheese. Transfer skillet to your preheated oven and cook for about 10 minutes until chicken is cooked through completely and cheeses are melted.
Spoon some of the onions and gravy over the top of the chicken, garnish with thyme or parsley and cracked black pepper, and serve.
If you don't have a cast iron skillet, any other oven-safe skillet will do OR you can transfer everything to a casserole dish or baking pan for the baking portion of the recipe.