Special equipment: a meat tenderizer
Combine 5 cups water and the turbinado sugar and ginger in a medium pot. Bring to a boil over medium heat, then turn off the heat and stir in the lime juice. Let cool to room temperature.
Pour the mixture into a 9-by-13-inch metal baking pan and place in the freezer. Freeze until completely frozen, about 4 hours or up to overnight.
Take the pan out of the freezer when it becomes a giant ice cube. Smash the frozen block with a meat tenderizer and scoop crushed granita into chilled martini glasses. Garnish with lime zest and serve immediately. Freeze any remaining granita until ready to serve.