Shred the cucumber into a bowl with a box grater. Add salt and vermouth, then stir and allow to stand for 10 minutes.
After 10 minutes, strain out the juices, pressing down on the cucumber pulp to get as much juice as possible. Discard pulp and reserve juices.
For 1 cocktail, put a tablespoon of the cuke juice in a chilled martini or coupe glass and swirl to coat. In a beaker, stir gin vigorously with ice until ice cold, then strain into the prepared glass. Garnish with a cucumber slice or a piece of cucumber peel, if desired, and enjoy immediately!