In a blender, combine the cucumber, jalapeño, 1 cup water, lime juice, and sugar, and blend until the mixture is smooth.
Strain the juice through a medium fine-mesh sieve into a pint container (discard the solids).
Add a pinch of salt, taste for balance, and add a bit more if necessary. Cover and refrigerate until ready to serve.
To serve, pour 2 oz. juice mix into an ice-filled Collins glass. Add 4 oz. club soda and gently stir. Garnish with a lime slice and serve. Repeat as desired.