Juice the cucumber. Put 1 cucumber through a juicer, or puree with the mezcal in a blender, then strain out the juice through a fine mesh strainer.

Prepare the glasses. Cut one of the extra limes into wedges and pour about 1/4 cup of the kosher salt on a small plate. Run the lime wedges around the edge of each rocks glass, then gently dip the edges in salt.

Stir together. Combine remaining ingredients, plus cucumber juice in a pitcher with ice and stir vigorously until everything is mixed and thoroughly chilled. Slice 1 of the limes into thin slices. Pour the margarita into salt-rimmed glasses and garnish with cucumber and lime slices and serve.