In a cocktail shaker, add the apples, curry powder, sugar and lime juice. Muddle well, until the texture resembles a chunky applesauce. Add the rum and set the shaker aside for 30 minutes to 1 hour.
Strain the liquid from the shaker into a separate container and discard the remaining pulp. Give the shaker a rinse, then add the mint and muddle. Add the strained apple-curry liquid to the mint, then fill the shaker with ice and shake well. Pour the liquid into a tall glass, fill to the top with more ice and top with the sparkling water. Insert a straw and serve.