Simmer water, fresh ginger, and ½ cup sugar in a small saucepan, uncovered, 10 minutes. Remove from heat and let steep 15 minutes. Strain syrup through a sieve into a bowl, discarding solids. Chill until cold.
Finely grind crystallized ginger with remaining ¼ cup sugar in a blender or food processor, then spread on a small plate.
Run lemon wedges around rims of glasses, then dip rims into ginger sugar. Put 1 tablespoon syrup into each glass and top off with Prosecco.
