Place the cucumber in an even layer on a baking sheet lined with parchment paper. Transfer it to the freezer. Pour the lime juice into an ice cube tray and transfer it to the freezer as well. Freeze both the cucumber and lime juice until solid. About 3 hours.
⅔ cucumber, 5 limes
When the cucumber and lime juice are nearly finished freezing, pour some salt in a sauce and a little lime juice in another. Dip four single rocks glasses first in the lime juice and then in the salt. Once the rims are evenly coated in salt, add a couple of ice cubes to each glass and set them aside. ****
Place the frozen cucumber and lime juice in a large blender. Add the ice cubes, mint, tequila, Cointreau, and agave. Blitz until smooth.
6-10 ice cubes **, ¼ cup fresh mint leaves, 8 oz tequila, 3 oz Cointreau, 1 oz agave syrup ***
Pour the mixture into a pitcher. Remove the ice cubes from the glasses and place a cucumber ribbon inside each glass. Fill the glasses with the margarita and garnish each with a sprig of mint. Serve immediately.
4 mini cucumber ribbons, 4 sprigs of fresh mint