Stir brown sugar, cornstarch, and salt together in a medium saucepan. Add 1/2 cup evaporated coconut milk; stir to combine. Add remaining evaporated coconut milk.
Heat over medium-low heat, stirring constantly, until mixture thickens to an almost pudding-like consistency, about 15 minutes.
Remove from the heat and stir in vegan butter and vanilla. Pour into one serving dish (or 4 smaller dishes) and cover with plastic wrap, so the wrap is touching the surface of the pudding.
Refrigerate for at least 4 hours before serving.