In a bowl, combine cocoa and both sugars, whisking to combine well. Add milk and whisk until sugars are dissolved. Stir in heavy cream and vanilla.
Pour mixture into the freezer bowl of ice cream maker. Let run for 30 minutes, or until mixture is thick, soft, and creamy. Transfer to airtight container and freeze at least 2 hours or until ice cream is to a desired consistency.
Prior to serving, let ice cream sit at room temp for 5 minutes to soften enough for easy scooping.