Brine the pork chops the night before: Combine 3 cups hot water, the vinegar, brown sugar, 1/4 cup salt, the garlic, sage and cayenne in a large bowl, stirring until the sugar and salt dissolve; let cool to room temperature. Submerge the pork chops in the brine, cover and refrigerate overnight.
Remove the pork from the brine and pat dry. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add half of the pork chops and cook until browned, 3 to 5 minutes per side (reduce the heat, if necessary). Transfer to a plate and repeat with the remaining 1 tablespoon olive oil and pork chops; transfer to the plate.
Sprinkle the flour over the drippings in the skillet and cook, stirring with a wooden spoon, until lightly golden, about 45 seconds. Add the chicken broth and scrape up any browned bits; switch to a whisk and cook, whisking, until the sauce begins to thicken, about 2 minutes. Add the milk, parsley, 1/2 teaspoon salt, and black pepper to taste and simmer 1 minute. Return the pork chops to the skillet along with any juices from the plate and heat through, about 2 minutes.