Place the pineapple juice and slices of ginger into a small saucepan. Bring to the boil, then simmer for 5-10 minutes until the liquid has reduced by half. Leave to cool to room temperature, then discard the ginger.
Make the caramelized pineapple next. Heat the butter in a frying pan until starting to foam. Dust the pineapple slices in the brown sugar and place in the frying pan. Fry, turning once, until both sides are golden and caramelized (about 3-4 minutes). Put to one side.
Place 6 mint leaves in each glass (this recipe is for two glasses). Add the lime juice and lime slices to the glasses.
Pour in the reduced pineapple-ginger juice and muddle everything together with a spoon.
Fill the glasses up two-thirds of the way with crushed ice, then pour over the spiced rum. Top with the club soda and mix again. Add more ice to fill up to the top if necessary.
Garnish the glasses with a small sprig of mint and a slice of caramelized pineapple.