To make Lemon Simple Syrup: Bring the sugar and water to a boil in a small saucepan over high heat, stirring constantly to dissolve the sugar. Remove from the heat and add the lemon juice. Let cool. (The syrup can be refrigerated in a covered container for up to 2 weeks.) Makes about 4 ounces (1/2 cup).
To make Beauty Elixir: Muddle the cucumber and strawberry puree well in a cocktail shaker. Add ice and gin, lemon juice, and lemon syrup; shake well. Strain into a chilled martini glass. Top with the sparkling wine. Garnish with a cucumber slice and serve.
Party-Sized Beauty Elixir: Pulse 8 slices cucumber and 3 ounces (6 tablespoons) strawberry puree in a blender to mince them. Add 6 ounces (3/4 cup) gin, 1 ounce (2 tablespoons) fresh lemon juice, and 1 ounce (2 tablespoons) lemon syrup and pulse just to combine. Strain the gin mixture through a wire sieve into a small pitcher. (The mixture can be covered and stored at room temperature for up to 3 hours.) For each drink, pour about 3 ounces (a generous 1/3 cup) into an ice-filled cocktail shaker. Strain into a martini glass, top with the sparkling wine, and garnish with a cucumber slice. Makes 4 cocktails.