In a medium saucepan, whisk together the dark brown sugar, unsweetened cacao powder, cornstarch, and a pinch of sea salt.
Gradually add the unsweetened oat milk to the dry mixture. Whisk continuously until everything is well combined and there are no visible clumps.
Place the saucepan over medium heat. Keep whisking often as the mixture heats up. The pudding will start to thicken as it approaches a gentle boil. This should take about 5-7 minutes.
Once the pudding has thickened, remove it from the heat and stir in the chopped dark chocolate and pure vanilla extract. Continue to stir until the chocolate has fully melted and is completely integrated into the pudding.
Pour the pudding into serving bowls and let it cool to room temperature. Once cooled, cover them with plastic wrap and refrigerate for at least 2 hours to set. Serve chilled, topped with your favorite toppings.