Combine water and sugar in a saucepan; bring to a boil, stirring until sugar is completely dissolved and syrup forms, about 5 minutes. Remove from heat, stir 1 cup mint leaves into mixture, and let steep for 30 minutes. Strain and discard leaves. Transfer syrup to refrigerator to cool completely.

Blend coconut, remaining mint leaves, lime juice, and lime zest together in a blender until mint is minced and mixture is well-combined, about 1 minute.

Stir coconut mixture, cooled syrup, and sparkling water together in a bowl.

Pour the mixture into the ice cream bowl, then select the ice cream mode to start making it.