In a medium bowl whisk the brine ingredients until the salt and sugar dissolve, 2 to 3 minutes.
Place the pork chops in a large, resealable plastic bag and pour in the brine. Press the air out of the bag and seal tightly. Turn the bag to distribute the brine, place in a bowl, and refrigerate for 1 to 1½ hours.
Remove the chops from the bag and discard the brine. Pat dry with paper towels (do not rinse) and season evenly on both sides with the rosemary and pepper. Let stand at room temperature for 15 to 30 minutes before grilling.
In a small saucepan over medium-high heat, boil the balsamic vinegar until slightly thickened and reduced to ¼ cup, about 10 minutes. Remove from the heat. Whisk in the butter and season with the salt, pepper, and sugar. Transfer to a small bowl.
In a small bowl mix the sauce ingredients.
Cook the potato slices in a large pot of boiling salted water just until they are cooked halfway through and beginning to soften slightly, about 5 minutes. Drain. Rinse with cold water to cool quickly. Drain again. Drizzle the potato slices with oil and season evenly with rosemary, salt, and pepper.
Prepare the grill for direct cooking over medium heat (350° to 450°F) and high heat (450° to 550°F).
Brush the cooking grates clean. Grill the chops over direct high heat and the potato slices over direct medium heat, with the lid closed as much as possible, until the chops are still slightly pink in the center and the potatoes are tender and grill marks appear, turning once or twice. The chops will take about 10 minutes and the potatoes will take 6 to 8 minutes. Remove from the grill as they are done. Place the potatoes in a medium bowl and toss with the parsley. Season with more salt and pepper, if desired. Place one pork chop on each of four plates.
Reheat the balsamic glaze. Divide the potatoes among the plates. Drizzle the chops with the balsamic glaze. Serve the sauce on the side as a dip for the potatoes.