1. In a medium saucepan over medium-low heat, bring 1 cup water, sugar, and ginger to a gentle simmer, stirring to dissolve sugar. You have now made a less-sweet gingered simple syrup. Remove simple syrup from heat, cover, and steep for about 30 minutes.

2. Over a medium bowl, pour syrup through a fine-mesh sieve, pressing minced ginger against sieve with the back of a spoon to extract flavor and the liquid. Discard ginger. Cover bowl and refrigerate until well chilled.

3. Evenly distribute peach slices among 6-8 tall glasses. For each glass, press peaches with a food mill masher, a muddler, or the back of a wooden spoon to release their juice. 

4. Stir the lime juice into the simple syrup. Pour about 2-3 tablespoons chilled syrup over peaches. Pour in seltzer until glasses are about half full, and stir to mix. Add ice cubes; then top with seltzer. Garnish with mint sprigs.

Store remaining syrup in refrigerator for up to 1 week.