Place a large oven-safe skillet over medium heat and add the butter.
Once the butter is melted, add onions to the skillet. Cook for 15 minutes, stirring often. The onions are ready when they turn a light caramel color and are nice and soft.
While the onions cook, place the chicken breasts between two pieces of plastic wrap. Pound the breasts to an even thickness. Season both sides of the breasts with thyme, salt, and pepper.
When the onions are done, remove them from the skillet and place them in a bowl.
Add the olive oil to your skillet. Once the oil is hot, added the seasoned chicken breasts to the pan. Cook for 5 minutes per side until cooked through. Remove the chicken to a plate.
Add the onions back to the skillet. Sprinkle flour over the onions. Stir and cook for 1-2 minutes to remove the raw flour taste.
Pour in the beef broth and stir to remove any lumps. Simmer until the sauce thickens.
Turn your oven on to broil. (see note)
Return the chicken to the skillet. Cover each chicken breast with 1 slice of provolone and 1 slice of mozzarella. Place the skillet under your oven's broiler and cook until the cheese is melted and bubbling.
Carefully remove from the oven. Serve warm and chopped parsley on top for garnish. (I like to spoon extra sauce over our chicken and put some on the mashed potatoes too.)
If you don't have an oven-safe skillet, carefully pour the sauce into a baking dish. Then add the chicken to the dish and cover with cheese before broiling.