Combine the cream, milk, sugar, cocoa powder, and salt in a small saucepan. Warm the mixture over medium heat, whisking frequently, just until the sugar and the cocoa dissolve. Remove from the heat and add the vanilla and peppermint extracts. Stir to combine and cool to room temperature.
When ready to churn, whisk the mixture again, and pour the mixture into the ice cream maker. Churn according to the manufacturer's instructions.
Transfer the finished ice cream to an airtight container and place it in the freezer until ready to serve. Serve with chocolate shavings on top, if desired.