Roughly chop half of the mint (stems and all) and set aside. Pick the leaves off the remaining mint and reserve separately.
Combine the sugar and 1/2 cup water in a small saucepan. Bring to a boil over medium heat, stirring occasionally to dissolve the sugar. Remove from the heat, add the chopped mint and half the cucumber slices and let cool.
Pour the cucumber-mint syrup through a strainer into a large pitcher, using the back of a ladle or rubber spatula to press down on the solids to extract as much flavor as possible. Add the lime juice, lime slices, remaining cucumber and picked mint leaves. Add enough ice to fill the pitcher (about 6 cups), then pour in the seltzer. Stir and serve immediately.