Preheat the oven to 400°F. degrees.

Heat a cast iron or another oven-safe skillet over medium-high heat.  Add butter to skillet and let melt. Add the sliced onions and stir around to coat with butter. Continue to cook, stirring often, for about 15 minutes or until the onions are translucent and softened.

Meanwhile, sprinkle chicken breasts with olive oil, salt, pepper, and thyme. Once onions are done cooking, remove onions from the skillet and set them aside.  Place chicken breasts in a pan and cook for 5 minutes per side.

Remove chicken from skillet and place on a plate. Cover with a paper towel to keep warm.  Return the onions to the skillet and sprinkle the cornstarch over the onions. Stir.

Pour in beef broth and use a whisk to stir the mixture and make sure the cornstarch is completely dissolved. 

Once the sauce is thickened, add the chicken back to the skillet and spoon sauce all over the chicken. Top one slice of provolone cheese and one slice of Swiss cheese over each chicken breast.

Transfer the oven-safe skillet to the preheated oven and cook for 8-10 minutes, or until the cheeses are melted and the chicken is cooked through. Spoon sauce and onions over the chicken. Garnish with chopped parsley.