How to Make It
Using a sharp paring knife, carefully cut out the center of each lime half, removing as much white pith as possible. Arrange lime halves, cut side up, in cups of a mini muffin pan to create a stable base for halves.
Combine sugar, water, basil, mint, and cucumber in a medium saucepan over medium heat. Bring to a simmer; cook, stirring constantly, until sugar dissolves completely. Remove from heat; let stand 5 to 7 minutes. Strain; cool syrup completely; discard solids.
Combine syrup, cucumber juice, and lime juice in a medium saucepan over medium-low heat. Sprinkle gelatin evenly over top; let soften 1 minute. Increase heat to medium; cook, stirring constantly, until gelatin dissolves completely. Remove pan from heat; stir in sake.
Pour mixture evenly into lime halves (about 2 1/2 tablespoons per half). Refrigerate 4 hours or until set. When ready to serve, remove from fridge; cut each lime half in half with a sharp knife to form 2 wedges. Place sanding sugar in a large shallow dish; dip each lime wedge into sugar to lightly coat filling.