Combine the sugar and 1/4 cup water in a small saucepan, set it over medium heat, and bring to a boil. Remove from the heat and whisk in the cocoa powder until the mixture is smooth and glossy, about 15 seconds. Let it cool to room temperature. Put the ice cream base and the chocolate mixture into a bowl and whisk to combine. Transfer it to the fridge to chill unil cold if you’re using a pre-frozen bowl.
Pour the mixture into an ice cream maker and turn on the machine. Churn just until it has the texture of soft-serve.
Quickly transfer the mixture to freezer-friendly containers: Spoon in a layer of ice cream, sprinkle on some brownies, and use a spoon to press them in gently. Repeat. Cover with parchment paper, pressing it to the surface of the ice cream so it adheres.