Cool a cup type glass with ice cubes. Pour all the ingredients into a shaker with some ice cubes. Shake and remove ice cubes from the glass. Serve in the glass by holding ice cubes in a colander.
In the metal bottom of a shaker, smash softly the basil leaves and the quarter of mango. Add the rest of the ingredients to the shaker and shake. Double strain to a martini type glass. Garnish with some mango slices.