1. Melt butter in medium saucepan. Add the rest of the sauce ingredients except the rum and cook over medium heat for 3-4 minutes. Stir in the rum, bring to a low boil and continue cooking for a few mintues to let the flavors blend. Remove from heat and strain out the ginger. Set aside
2. Slice the pineapple in half lengthwise, including the top/leaves in the cut. Remove the core. Carefully cut the fruit from the skin as much as you can with a sharp, narrow knife, keeping the outer pineapple intact. Slice the pineapple into 1/2" slices, again being careful not to cut all the way through. Some of the fruit will still be anchored to the bottom of the pineapple "bowls".
3. Pour the warm Butter-Rum sauce over the fruit. Grill the pineapple bowls on a medium to medium-hot grill until the fruit is hot and tender, about 20-30 minutes, basting occasionally. The bowls can then sit on a warm spot of the grill until you're ready for them. Finish cutting the fruit from the bowls and serve.