First, make the cucumber juice. Dump the cuke chunks into a blender and blend until liquefied (you may have to pause between pulses to push the larger pieces down with a spatula).
Set the pulp in a fine mesh strainer over a bowl. Use a spatula to help the juice through. I liked mine a little rustic (and the thicker juice gave the end result a great texture, so once through the strainer was perfect for me!). For a clearer drink, double strain the juice. Either way, you should get between 1/2 and 2/3 cup juice.
Pour all of the liquid ingredients (tequila, triple sec, and cucumber juice) into a glass. Stir in sugar until dissolved.
Run the lime wedge along the rim of a fresh glass to dampen it. Dump kosher salt into a shallow dish. Invert the glass into the salt dish and spin to adhere salt to rim.
Fill the salted glass with ice. Add drink and serve.