In a small pot, bring the water, salt, garlic, sage, and peppercorns to a boil. Remove from heat and stir in apple cider. Let the brine cool to room temperature.
Place the pork chops in a glass baking dish or bowl (you could also use a sealed bag). Pour the cooled brine over the top of the pork chops, making sure they're completely covered, and let them brine for 1 to 2 hours (or up to 12 hours) in the fridge.
Remove the pork chops, give them a rinse with cold water, and pat dry with paper towels. Let them come to room temperature for 10 to 15 minutes.
Heat a large cast-iron skillet over medium-high heat. Add 1 tablespoon of butter along with the oil. Once shimmering, turn the heat down to medium and add the pork chops. Sear for 3 to 4 minutes on each side, until golden brown. Remove the chops from the pan, set on a plate, and cover with foil to keep warm.
Reduce the heat to medium, add another 2 tablespoons of butter to the pan, along with the coconut sugar, and stir together. Add the sliced apples and cook for 2 to 3 minutes, until softened but still firm. Add the shallot and chopped sage leaves, and stir for another minute.
Add the apple cider and mustard to the pan, and gently stir together. Use tongs to add the pork chops back to the pan, and bring the liquid in the pan to a simmer while warming the pork chops, about 2 to 3 minutes. Spoon the sauce over the pork chops and use an instant read thermometer to ensure the pork chops are cooked through to 145°F.