In a medium pot, melt the butter over medium low heat. Add the brown sugar and stir until it forms a glossy, caramel-like paste. Allow it to bubble gently for about a minute.

Slowly whisk in the milk and cream, warming the mixture until it is hot but not boiling.

In a separate bowl, whisk the egg yolks, cornstarch, and a pinch of salt until smooth. Gradually pour a bit of the hot milk mixture into the egg bowl while constantly whisking to temper the eggs.

Pour the egg mixture back into the pot and whisk constantly over medium heat until the pudding thickens and coats the back of a spoon.

Remove from heat and stir in the vanilla extract. Pour into cups or a bowl, pressing plastic wrap directly onto the surface to prevent a skin from forming. Chill for at least 2 hours.