In a large saucepan, combine the brown sugar, cocoa, cornstarch, and salt, whisking lightly. Add in the milk and cream to the sugar mixture and whisk until it is smooth. Stir in the chocolate chips and butter and bring the mixture to a simmer over medium heat, whisking or stirring occasionally. Once the mixture is at a simmer, reduce the heat to medium-low and cook, whisking constantly, until the mixture is thickened and large bubbles pop at the surface, 2 to 3 minutes. Remove pot from the heat and stir in the vanilla.

Transfer the pudding to a large bowl (or small serving dishes) and cover with plastic wrap, pressing the plastic wrap directly onto the surface of the pudding to avoid a skin starting to form. Refrigerate the pudding until it is completely cool, at least 4 hours but up to 3 days, if it is stored in an airtight container. Serve with whipped cream if desired.