In a large bowl, whisk together cocoa powder, granulated sugar, light brown sugar, and kosher salt until well combined.
Gradually add the whole milk to the dry ingredients, whisking continuously until the mixture is smooth and the sugars have dissolved. Stir in the heavy cream and vanilla extract until just combined.
Cover the bowl with plastic wrap and refrigerate for at least 2 hours, or overnight, to chill thoroughly.
Once chilled, pour the mixture into an ice cream maker and churn according to the manufacturer's instructions until it reaches a soft-serve consistency, typically about 15-30 minutes.
Transfer the churned ice cream to a freezer-safe container, cover tightly, and freeze for at least 2 hours to firm up before serving.