To make the syrup, combine water, sugar and ginger in a saucepan over medium low heat. Stir and cook until the sugar has completely dissolved. Remove from heat and steep at room temperature for about 30 minutes. Pour the syrup through a fine mesh sieve, using the back of a wooden spoon to press all the syrup out of the ginger. Place in a covered jar and refrigerate.
Evenly distribute peach slices into 4 tall glasses, (or 2 tablespoons peach puree) smash the peaches with a spoon to release their juice. Pour about 2 tablespoon ginger syrup over the peaches in each glass. (The remaining syrup can be stored for up to 2 weeks in the refrigerator.) Add ice cubes and then top off with sparkling water. Float a few mint leaves on top and serve.