Fill a cocktail shaker with crushed ice and add the vodka and lime juice as well as two of the ginger slices (if desired). Shake well and strain into a copper mug. If you don't have a copper mug, you can use a highball glass. Add the ice from the shaker to the mug.
Pour in the ginger beer, add a straw, lime wedge and remaining ginger slices as garnish. Slap the sprig of mint between your hands and place in the drink next to the straw.
For a virgin version, omit the vodka and mix directly in the mug.