Muddle fresh ginger and mint leaves in a cocktail shaker with a bit of ice. Add carrot juice, vodka, ginger liqueur, simple syrup and lemon juice. Shake well.
Pour into tall glass filled with ice. Top with some ginger beer.
Notes
If I have mint simple syrup on hand I will use that and not muddle fresh mint. You can also replace the fresh ginger with about 1/2 tsp. minced ginger paste.