1. Combine 1/4 cup water and sugar in a small saucepan over medium-high heat; cook until sugar dissolves. Remove from heat. Add ginger. Cover and let stand 15 minutes. Strain sugar mixture through a fine sieve. Discard solids. Chill sugar mixture 30 minutes or until ready to use.
2. Fill a large martini shaker half full with ice. Add sugar mixture, juice, and cognac; shake until chilled. Strain 1/2 cup mixture into each of 10 martini glasses. Garnish each serving with 1 lemon twist.