1. Prepare the Crust:

1. Whisk brown sugar, cornstarch, and salt in a medium saucepan. Gradually whisk in milk until smooth.
2. Cook over medium heat, stirring constantly, until mixture thickens and begins to bubble, about 10-12 minutes.
3. Whisk egg yolks in a small bowl. Gradually whisk in 1 cup of the hot milk mixture to temper the eggs.
4. Pour the egg mixture back into the saucepan and cook for 2 more minutes, stirring constantly.
5. Remove from heat and stir in butter and vanilla until fully incorporated.
6. Pour pudding into serving dishes and press plastic wrap directly onto the surface. Chill for at least 4 hours before serving.