In a medium saucepan, combine brown sugar, cornstarch, and salt. Whisk together until well mixed and no lumps remain.

Gradually add milk while whisking constantly to create a smooth mixture without lumps.

Place the saucepan over medium heat and cook, stirring constantly with a wooden spoon or whisk, until the mixture begins to thicken and bubble, about 8-10 minutes.

Once thickened and bubbling, continue cooking for 2 more minutes, stirring continuously to prevent scorching on the bottom.

Remove from heat and immediately stir in butter, vanilla extract, and a pinch of salt. Whisk vigorously until the butter is completely melted and incorporated.

In a small bowl, lightly beat the egg yolks. Slowly add about half cup of the hot pudding mixture to the egg yolks while whisking constantly to temper them.

Pour the tempered egg mixture back into the saucepan with the remaining pudding, whisking constantly.

Return the saucepan to medium-low heat and cook for 2-3 minutes more, stirring constantly, until the pudding reaches a thick, creamy consistency that coats the back of a spoon.

Remove from heat and strain the pudding through a fine-mesh sieve into a clean bowl to ensure a perfectly smooth texture.

Divide the pudding among six serving dishes or ramekins. If you prefer no skin on top, press plastic wrap directly onto the surface of each pudding.

Refrigerate for at least 2 hours or until completely chilled and set before serving.