1. Combine fresh ginger and sugar in bowl of a food processor. Process until ginger is finely ground. Transfer to a large saucepan, add lemon juice and stir in 2 cups cold water. Bring to a boil over high heat, stirring. Reduce heat slightly and simmer until thick and syrupy, about 30 minutes. Remove from heat, cover and steep for 20 minutes. Strain and cool. (You should have about 2 cups.) Syrup can be made up to 2 weeks in advance. Keep tightly covered and refrigerated.
2. Just before serving, fill 8 tall glasses with crushed ice. Pour about 1/4 cup of chilled ginger syrup into each glass. Fill with chilled seltzer and serve right away, garnished with lemon slices, if desired.