To cut chops from a bone-in pork loin, follow along the rib bone with a sharp knife to make one clean cut. Season the pork chops thoroughly with kosher salt and fresh ground pepper. One at a time, cover each chop with a piece of plastic wrap and lightly pound with the back of a heavy knife or a spoon to work the seasoning into the chops.Cut the peaches into medium ½-inch chunks. Slice the garlic cloves into thin slivers. Pull the sage leaves from the stems and thinly slice.
Heat a heavy bottomed cast-iron pan over high heat. Once the pan is hot, add 2 tablespoons of the olive oil.
To the hot pan, add all of the chops standing up on their fat cap side, to render off some of the fat, and pan-sear for a few minutes. Remove two of the chops, and lay the other two down on their meatier sides, to pan sear and cook through. After a few minutes, carefully turn over and brown the other side for another few minutes. Remove from the pan and set aside to rest.
Repeat for the remaining chops. Remove from the pan and let rest with the other pork chops.Drain and discard some of the cooking oil. Add the peaches to the pan and sauté vigorously, about 2 minutes. Add the garlic and sage, and immediately add the apple cider. Season the mixture with salt and freshly ground black pepper. Add the butter and swirl the mixture together, to emulsify the butter into the peaches. Remove from the heat.
Stack the rested pork chops on a platter and finish by spooning the peaches and pan juices over top. Serve immediately.