Drain the potatoes. Heat 2 tablespoons of the olive oil in a 12-inch cast-iron skillet over high heat until very hot, about 2 minutes. Add the potatoes and cook, stirring occasionally, until they start to become tender and are just beginning to brown around the edges, about 5 minutes.
Sprinkle the pork chops with salt and pepper. Move the potatoes to the far edge of the pan, leaving a space to brown the pork chops. Add the pork chops to the pan and cook until browned, 2 to 3 minutes per side. As the pork chops cook, give the potatoes an occasional stir so they continue to brown evenly. Place the pork chops on top of the potatoes, shingling them to leave as much room in the pan as possible. Reduce the heat to low and add the remaining tablespoon of oil to the bare area of the pan. Add the olives, capers, garlic and thyme and cook, stirring continuously, until fragrant and golden, 1 to 2 minutes.
Increase the heat to medium, add the tomatoes and wine and cook until reduced by half, 2 to 3 minutes, then stir in the chicken broth. Put the pork chops in the sauce and carefully nestle the potatoes around them. Cook 3 to 5 minutes more until the pork chops register 145 degrees F in the center on an instant-read thermometer. Remove the pork chops from the sauce and transfer to shallow bowls or a serving platter. Taste the sauce and season with additional salt and pepper if needed. If most of the liquid in the pan evaporates while you are cooking the pork, stir in tablespoons of water at a time to get it back to a saucy consistency. If the sauce is a little thin and weak, after you take the chops out, turn the heat up and cook 1 to 2 minutes more to thicken and concentrate the flavors. Stir the parsley into the sauce, remove the thyme sprigs, spoon the sauce over the chops and serve.
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