Make the agua fresca: Strip the leaves from the mint stalks. In a blender, combine the mint, cucumber, lime juice, sugar and 2 cups of water. Blend on high until smooth. Look at the measurements on the side of your blender jar and add enough water to bring the quantity to 4 cups. Strain to remove any seeds or unblended bits. Pour into a storage container, cover and refrigerate until you are ready to use—which should be within a few hours for the freshest flavor and look.
From agua fresca to cocktail: In the bottom of a large (2-quart) pitcher, muddle the ginger with a cocktail muddler or wooden spoon until you have a coarse mash. Add the tequila, Ginger Liqueur and Tangy Cucumber-Mint Agua Fresca. Stir, cover and refrigerate until chilled, about 2 hours.
Fill 8 12-ounce highball glasses with about a cup of ice each. Pour in a generous ¾ cup of the cocktail mixture, making sure to get some of the muddled ginger in each glass. Top each glass with an ounce of soda. Stir with a long-handled bar spoon, garnish with a lime wedge and carry to your guests.