Step 1
Bring 4 cups water to a simmer in a medium saucepan. Remove from heat and add 4 mint tea bags and ⅓ cup sugar; let sit 5 minutes. Using a slotted spoon, remove tea bags, gently pressing with a spoon to extract as much liquid as possible. Stir to dissolve sugar.
Step 2
Pour tea mixture into a large pitcher and add 3 Persian cucumbers, thinly sliced, pressing down to submerge. Let cool (about 20 minutes).
Step 3
Add 1 cup plus 2 Tbsp. fino or manzanilla sherry, ¼ cup plus 2 Tbsp. fresh lime juice, and ¼ cup plus 2 Tbsp. gin to tea mixture; stir punch to combine.
Step 4
To serve, pour one 1-liter bottle chilled club soda into rocks glasses filled with ice to come halfway up sides. Top off with punch. Garnish with large mint sprigs.