Kiwi Snow Kone

Bring sugar and water to a boil in a small saucepot over medium-high heat. Cook until sugar is dissolved and liquid is slightly reduced, about 5 minutes. Remove from heat and cool. Transfer to a blender, add the kiwi and puree until smooth. Pour into a pitcher and stir in lime juice. Keep cool. Finely shave ice in a food processor and divide among 4 cups. Pour kiwi mixture over ice and serve immediately.