Season pork chops with salt and pepper. Let them come to room temperature.
Preheat the oven to 400F degrees. Place cast iron skillet in oven.
Heat the butter in a saucepan over medium heat and add the onions. Cook (stirring periodically) until the onions are golden brown and starting to caramelize, about 15-20 minutes.
While the onions cook, remove the heated cast iron skillet from oven and place over medium high heat.
Add the pork chops and sear until golden brown (about 2-3 minutes per side, depending on thickness).
Add the sherry, broth and thyme to the pan. Top with caramelized onions.
Place in the oven and cook for about 5-7 minutes. The internal temperature should be about 145 degrees.
Remove from oven and turn the broiler on.
Lay 1-2 slices of cheese over each pork chop.
Place under the broiler for 1-2 minutes, or until cheese starts the lightly brown.
Drizzle sauce over the pork chops and serve immediately.