Bring the ginger, cardamom seeds, and 1¼ cups water to a boil in a saucepan. When the mixture boils, cover the saucepan and reduce the heat to a low simmer. Let the liquid simmer for about 30 minutes.

Strain the liquid into a saucepan through a strainer lined with cheesecloth. Discard the solids. You should have about 1 cup of liquid. Bring the liquid and the sugar to a boil, stirring constantly to dissolve the sugar. When the sugar is dissolved, remove the pan from the heat and let it cool to room temperature. When the liquid is cool, put it in a covered container or squeeze bottle and refrigerate until cold.

Peel the peaches and remove the pits. Purée the flesh in a blender, then strain the juice through a fine-mesh strainer or food mill. You should end up with about 1 cup of juice. Discard the solids and store the juice, covered, in the refrigerator.

To make each drink, stir together 3–4 tablespoons peach juice, depending on how much peach flavor you want, and ¼ cup chilled ginger-cardamom syrup. Fill a tall glass with ice and pour the mixture over the ice, followed by 1 cup club soda. Stir gently with an iced-tea spoon to combine. Garnish with a mint sprig or lime wedge, if desired.