In a small saucepan, bring the half-and-half, water, and sugar to a simmer. Remove from the heat and whisk in the cocoa, malt powder, and vanilla extract until completely dissolved. Strain, then set aside to cool.
Transfer to an ice cream maker and freeze according to the manufacturer's instructions. Put the finished frozen egg cream in a storage container and freeze until firm.
Serve garnished with whipped cream and mint sprigs.