Combine the brown sugar, cornstarch and salt in a medium-sized saucepan and whisk until well combined. Continue to whisk, slowly adding the cream.
Set the pan over medium heat and slowly bring to a simmer, whisking and stirring often, scraping around the edges with a high-temperature rubber spatula. Once at a simmer, whisk constantly until thickened, making sure to scrape the sides of the pan. This should prevent lumps.
When thickened enough that the whisk leaves a deep trail in the pudding, remove from heat and whisk in the butter, 1 tablespoon at a time. Whisk in the vanilla.
Transfer to 8 custard cups (or 1 (8 cup) dish) and refrigerate overnight (or up to 3 days). Serve cold with whipped cream.