In a large bowl, whisk the brown sugar and the ½ cup of salt with 3 cups of water until dissolved. Add the pork chops and refrigerate for at least 30 minutes and up to 12 hours.
Meanwhile, in the bowl of a stand mixer fitted with the paddle attachment, beat the butter with the milk at medium speed until light and fluffy. Add the jam and ¼ teaspoon of salt and beat until incorporated and the butter is smooth. Be careful not to overbeat your butter or the milk solids will separate. Scrape into a small container and set aside at room temperature.
Preheat your oven to 400°F.
Drain the pork chops and pat dry. Preheat a large cast iron skillet over moderately-high heat until very hot. Add the oil. Add the pork chops and cook until the underside is golden brown, about 3 minutes. Flip the chops, top each with a spoonful of the apple butter and 2 sage leaves.
Carefully transfer the skillet to the oven and cook until the chops are just cooked through and a meat thermometer inserted in the thickest part reads 140°F, 6 to 8 minutes depending on the thickness of your chops. Transfer the pork chops to a plate and let rest for at least 10 minutes.
Divide the potato puree between two dinner plates. Top each plate with one of the pork chops and some sautéed green beans. Top each chop with another spoonful of the apple maple bacon butter. Serve right away.