Make the rub: Combine the brown sugar, salt, coriander, black pepper, thyme, allspice, cinnamon, and cayenne in a small bowl.
Rince the chicken breasts and pat dry with paper towels. Place in a baking dish and drizzle both sides with the sesame oil. Sprinkle the rub on both sides of the chicken, followed by the garlic, scallion white, and ginger, patting the seasonings onto the meat with your fingers. Cover the chicken and let it cure in the refrigerator for 1 hour.
Make the Pineapple-Rum Glaze: Combine the butter, brown sugar, rum and pineapple juice in a saucepan over high heat. Bring to a boil and boil until syrupy, 5-8 minutes. Set aside.
Grill the chicken breasts for 2 minutes, rotate a quarter turn and grill 2 more minutes. Turn them over and brush with some of the glaze. Grill until cooked through, approximately 4-6 more minutes, rotating a quarter turn after the first 2 minutes. Turn over again and brush with glaze and grill for 1 minute longer. Total cooking time is 9-11 minutes depending on thickness of chicken breasts.
Transfer the chicken breasts to a platter and drizzle with any remaining glaze. Sprinkle scallion greens on top and serve.