A 3 3/4-pound pineapple yields about 3 cups of fruit. When you core and cut it, save the juice.
Put pineapple, any juice, and ginger in a blender or food processor. Whirl until fruit is coarsely puréed. Add scoops of ice cream and whirl until smooth. Add rum and whirl to blend. Fill a fresh pineapple shell or tall glasses with milkshake; add straws for sipping. Refill pineapple shell as needed with remaining shake.