Set up your barbecue for direct cooking. Please refer to your handbook for further instructions. Preheat your barbecue for barbecuing.
Once the barbecue has preheated, adjust the burner(s) to medium. Place the pineapple on the barbecue. Barbecue for 3 minutes per side.
Place the lime halves on the grill and cook for one minute. Remove the pineapple and lime halves from the barbecue and set aside.
Place the lime halves, cut side down in the brown sugar.
Divide the mint leaves between two tall glasses. Squeeze the juice from the limes into the two glasses. Muddle the mint leaves with a muddler or another similar object. Pour the crushed ice into the two glasses. Pour in 45 ml of white rum in each glass and top with soda water. Stir for a couple seconds. Add the slices of grilled pineapple
Finish off with a garnish of mint leaves.