With a thin-blade knife (or kitchen mandoline), cut 12 thin round slices of cucumber (with the peel). Cover and refrigerate
Peel the remaining cucumber and cut into chunks. In the container of a blender, add the cucumber, elderflower liqueur, and lime juice, and blend on high until completely puréed.
For each drink, add ice to a cocktail shaker along with a jigger of vodka. Add 2 jiggers of the cucumber purée, and shake vigorously. Strain the contents into a chilled martini glass and garnish with 3 cucumber slices. Serve immediately. Repeat for additional cocktails.
Notes
You can use the regular cucumbers or the English variety. You can remove the seeds, but I find that my heavy-duty blender will pulverize them fine. If you do not have a blender, then muddle the cucumbers (definitely remove the seeds) in a cocktail shaker until puréed. Although strained at serving, some of the pulp of the cucumber is a key component of the drink and will coat the inside rim of the glass for effect. I will admit that buying cucumber-flavored vodka is a nice touch, but not necessary for the recipe; feel free to use any quality unflavored vodka. Optional to the martini glass, mix up a big batch of the cocktail (vodka-added) in a pitcher, pour over ice-filled glasses, and garnish with cucumber slices.
Recipe by Acadiana Table at https://acadianatable.com/2023/08/21/cucumber-martini/