Combine jerk seasoning, 1 tablespoon lime juice, and 1 teaspoon sunflower oil in a medium bowl. Rub all over the pork chops and allow to rest about 20 minutes.
Cut the mangoes vertically, so you will have two even pieces, leaving the pit. Cut each mango piece into 1-inch cubes, cutting to, but not through, the skin. Brush cut sides of mangoes with the remaining 1/2 teaspoon oil.
Heat a grill pan over high heat, and rub lightly with sunflower oil. Cook the pork for 4 minutes on each side. Cook the mangoes, skin sides up, for about 3 minutes. Remove from heat.
Combine brown sugar, rum, butter, and salt in a small saucepan. Bring to a boil, then reduce heat and simmer until thickened, about 4 minutes. Stir in 1 tablespoon lime juice. Drizzle the sauce over the pork chops and mangoes and serve.