3/4 cup water
1/2 cup sliced fresh ginger (2 ounces)
3/4 cup sugar, divided
1 tablespoon finely chopped crystallized ginger
2 lemon wedges
2 (750-ml) bottles chilled Prosecco
Simmer water, fresh ginger, and 1/2 cup sugar in a small saucepan, uncovered, for 10 minutes.
Remove from heat and let steep 15 minutes.
Strain syrup through a sieve into a bowl, discarding solids and chill until cold.
Finely grind crystallized ginger with remaining 1/4 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.