Combine 2 1/2 cups water, brown sugar, granulated sugar, and cocoa in a saucepan. Bring to a boil over medium-high heat and cook, whisking occasionally, until the sugar dissolves, 4 or 5 minutes. Reduce the heat to low; boil gently for 3 minutes.
Meanwhile, set a medium bowl in an ice water bath. Remove the syrup from the heat and add the chocolate and vanilla; whisk until the chocolate is melted and well incorporated. Pour the mixture into the bowl over the ice water bath and stir occasionally until well chilled. Transfer to an ice cream maker and freeze according to the manufacturer's instructions.