Combine vodka and shiso in a large bowl. Cover, and let stand overnight at room temperature. Strain into a pitcher, and discard shiso.
Add vermouth and lime juice to vodka. Fill a cocktail shaker halfway with ice. Add 1/2 cup vodka mixture, and shake for 45 seconds. Pour into a chilled martini glass, and garnish with a shiso leaf. Repeat, and serve immediately.
Shiso is a delicate Asian herb with tender, slightly fuzzy leaves and an almost minty flavor. Gin may be substituted for vodka.