Make the brine. In a measuring cup, mix together the warm water, salt, and brown sugar and stir until the sugar and salt are dissolved. Stir in the apple cider and let the brine cool to room temperature. Put the pork chops in a large ziplock bag and pour in the brine. Carefully press as much of the air out of the bag as possible, to make sure the chops are submerged, and seal the bag. Refrigerate for at least 2 hours, up to 24.
Preheat an oven to 400 F.
Sear the pork chops. Heat a large cast iron skillet or braiser (12-13”) over medium heat for 5 minutes. Add the oil and swirl it around the pan. Lay the pork chops in a single layer in the skillet and sear for 3-5 minutes, until deeply golden brown on one side. You may need to press the pork into the pan to keep the meat from bowing up. Flip the chops and turn off the heat.
Baste the pork chops. Scoot all the pork chops to the side of the pan furthest from you and add the butter, garlic, maple, and chopped sage to the open space in the pan closest to you. Once the butter has melted, use one hand to tilt the pan towards you and use a spoon to baste the maple/sage butter all over the pork chops for 2 minutes. Transfer the pan to the oven for 8-10 minutes, or until the pork chops reach an internal temperature of 150 F.
Make the pan sauce. Remove the pan from the oven and transfer the pork chops to a plate to rest. Bring the pan drippings to a simmer and stir in the dijon mustard and flour. Let the flour cook for 2 minutes, while stirring. Whisk in the apple cider and simmer for 3-4 minutes, until thickened. Season to taste with salt and pepper. Spoon the pan sauce over the pork chops and serve right away.