Make the cucumber juice. Blend the cucumber in a blender on high for 30-40 seconds until it looks like thick juice. A juicer will also work for this step. Strain the pulp through a fine-mesh strainer, using the back of the spoon to press the cucumber solids to release more juice.
Fill a cocktail shaker with ice and add the tequila, 6 ounces of the cucumber juice, agave, and lime juice.
Shake all ingredients over ice.
Salt the rim of two cocktail glasses and fill with ice.