Preheat oven to bake at 400 degrees OR broil on line. In a large oven-safe skillet (see note) over medium-high heat, melt butter. Add onions and 3 tablespoons beef broth and sauté 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.

Pat chicken dry with paper towels, season with salt, pepper, 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 golden brown on both sides.

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 frequently, for 1-2 minutes until slightly thickened. Taste and adjust 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 ¼ of the parmesan cheese. Transfer skillet to preheated oven and cook for about 10 minutes until chicken is cooked through completely and cheese is 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.