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.
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.
Remove the ice cubes from the glasses and place a cucumber ribbon inside each glass. Fill the glasses with the margarita and garnish with a sprig of mint.