Coat the rim of a rocks glass with the dill salt.
Add all the ingredients to a shaker with ice, and roll the mixture back and forth with another shaker 3 times.
Strain the cocktail into the rimmed glass over ice.
Garnish with a lemon wheel.
*Cucumber juice: Either use a juicer or purée cucumber in a high-powered blender, then strain through a fine-mesh sieve.
**Dill salt: Stir together 1 sprig of dill and 1/4 cup salt in a bowl, and let sit for at least 2 hours before using. Makes 1/4 cup, enough to rim 12 glasses.