In a saucepan, combine brown sugar, cornstarch, and salt.
Gradually whisk in milk until smooth.
Cook over medium heat, stirring constantly, until the mixture thickens and comes to a boil. Boil for 1 minute.
Remove from heat and stir in butter and vanilla extract.
Pour into a mixing bowl and let cool slightly. Cover with plastic wrap, pressing it directly onto the surface of the pudding to prevent a skin from forming.