Combine water, sugar, and cinnamon stick in a saucepan.

Bring to a boil, then reduce heat and simmer for 5 minutes, stirring until sugar dissolves.

Strain the syrup and refrigerate until cold.

Puree blueberries in a blender or food processor until smooth.

Strain the blueberry puree and refrigerate until chilled.

Combine the chilled syrup and blueberry puree.

Pour the mixture into a sorbet machine and process as directed.

Transfer the sorbet to the freezer and let it harden for about 6 hours.

Shave the frozen sorbet into tiny slivers using a sharp knife.

Serve the shaved ice in chilled glasses.