Brine the chops: Combine water, brown sugar, salt, peppercorns, and bay leaves in a large pot and bring to a boil. Simmer for 20 minutes, then give it a good stir to make sure all the salt and sugar are dissolved. In a 3-gallon food-safe bucket, add the brine and enough ice until you have exactly 1½ gallons of brine. Once the brine is cold, place the pork chops in brine, cover, and refrigerate for 24 hours. Remove chops from brine and pat dry with paper towels. Season heavily with more salt and fresh-ground black pepper.
Make the gastrique: In a saucepan, combine cane syrup and cane vinegar. Bring to a boil over medium heat and cook until the liquid has reduced to 1 cup, about 20 minutes. There’s no need to stir, but watch closely, as it likes to burn. You can make this in larger batches, and the shelf life is pretty much infinite. Store in a sealed jar (does not need refrigeration).
Grill the chops: Preheat grill to high. Preheat oven to 400°F. Grill pork chop (even bone side) for 2 to 3 minutes on each side to get really hard grill marks. Put chops in a roasting pan and top each with 2 tbsp. butter. Roast for 8 to 10 minutes, until it reaches at least 145°F internal temperature. Allow chops to rest for 3 minutes in the pan. Before serving, dip them on all sides in the juices and butter that remain in the pan.
To serve: Place pork chops on top of dirty rice, and drizzle ¼ cup gastrique over each. Garnish with green onions.