Pour kvass into a small saucepan and boil over medium heat for 5 minutes, cool.
Pour ice cubes into a mixing glass, pour kvass, add vodka, stir with a cocktail spoon.
Separately, in a bowl, mix currant syrup and cream, beat with a whisk until thick.
Strain the cocktail into wide martini glasses, top the cocktail with a layer of whipped cream. Garnish with mint leaves and currants.