First prepare the peppers. Heat a little olive oil in a large frying pan, then add the onion and peppers. Season with salt and pepper, add the sugar, and sauté over high heat for 4-5 minutes until soft and colored. (Make sure you can hear the vegetables hissing in the pan. If not, the pan isn't hot enough and you're in danger of boiling the vegetables instead of sautéing them.)
Add the vinegar and let it bubble for a minute or two until it has reduced and the peppers are soft. Turn down the heat, add the tablespoon of extra virgin olive oil, and cook for additional 2-3 minutes. Stir in the shredded basil and continue to cook for 30 seconds, then turn off the heat. Transfer to a bowl and set aside to infuse. Wipe the pan clean, ready to cook the pork.
Using a sharp knife, make cuts into the fat of the chops, about ¼ inch deep and at 1½ inch intervals, making sure you don't cut into the meat. (This will stop the meat from curling up during cooking and will make it cook more evenly.) Season the chops with salt and pepper really well on both sides, pushing the seasoning into the meat.
Place the cleaned-out frying pan over high heat until hot and add a dash of oil. Add the chops, garlic, and thyme and cook for 2-3 minutes until colored. Turn and cook for another 2-3 minutes on the other side, pushing the thyme under the chops and breaking up the garlic a little.
Toward the end of cooking time, add 3 chunks of butter and baste the chops with it as they are cooking, to speed up the cooking process and keep the chops moist. (Push the fatty edge of the chops toward the back of the pan to help render the fat.) Squeeze the garlic out of its skin and place it with the herbs on top of the chops.
Transfer the chops to a plate, and let rest for 5-10 minutes, spooning over the basting butter now and again.
Serve the chops on top of the peppers with the resting juices and a little juice from the peppers.