Heat 1 Tbs. of the oil in a very large sauté pan over medium heat. Add the onions and cook, stirring frequently, until soft and caramelized, about 20 minutes (if the onions start to burn, reduce the heat to medium low). Transfer the onions to a bowl and set aside.
Pat the pork chops dry with paper towels and rub both sides with the salt and pepper. Return the pan to the burner, add the remaining 2 Tbs. olive oil, and increase the heat to medium high until the oil is hot but not smoking. (If you don’t have a sauté pan large enough to fit the chops without crowding, use two smaller sauté pans, heat 1-1/2 Tbs. olive oil in each, and cook two chops in each.) Cook the chops until well browned on one side, 3 to 5 minutes. Turn and cook the other side until the meat is done, 2 to 4 minutes. (If they start to burn, turn down the heat slightly.) To check for doneness, make a small cut near the bone and look inside—the pork should have a hint of pinkness. If it’s still red, cook for another minute and check again. Transfer the chops to a plate, tent with foil, and let rest for 3 to 5 minutes before serving.
Meanwhile, put the pan back over medium heat and add the balsamic vinegar and caramelized onions (if you used two pans for the chops, use just one for the onions). Simmer, scraping the pan with a wooden spoon, until the vinegar is reduced enough to just coat the onions, 2 to 3 minutes. Stir in the thyme and season with salt and pepper to taste. Serve immediately with the pork chops.