Heat a large skillet to medium high heat. Season the pork chops on both sides with salt and pepper.
Once the skillet is hot, add in 2 tbsp butter, then add pork chops to the skillet. Cook for 4-5 minutes per side, or until browned. It's okay for them to not be cooked all the way. Remove the pork chops from the pan and place on a plate and cover with foil. Keep the skillet hot!
Add the remaining butter to the skillet and let the butter get golden brown (about 1 minute). Turn heat down to medium. Saute the onion and apples for 3-4 minutes, then add garlic and 1/2 the sage, saute for another minute. Sprinkle arrowroot starch on the onion/apple mixture.
Pour in the apple cider and whisk in the dijon mustard. Turn the heat up to medium high and let the sauce thicken (about 1-2 minutes). Place pork chops back into the pan and let cook another 2-3 minutes.
Serve pork chops hot and spoon sauce over each pork chop. Garnish with remaining sage, season with salt and pepper.