In a medium saucepan, combine 375ml of the milk with the sugar and butter. Bring to a gentle simmer over medium heat, stirring occasionally, until the butter is completely melted and the sugar is dissolved.
In a separate bowl, whisk together the egg yolks, the remaining 125ml of cold milk and the entire packet of vanilla pudding powder until completely smooth with no lumps. This mixture must be completely smooth before it goes into the hot milk. Any lumps at this stage will remain as lumps in the finished cream.
Add the vanilla extract to the egg yolk mixture and whisk through.
Now temper the egg mixture: take a ladleful of the hot milk from the saucepan and pour it slowly into the egg yolk mixture, whisking continuously as you pour. This gradually raises the temperature of the egg mixture without scrambling it. Add a second ladleful and whisk again. Now pour the entire tempered egg mixture back into the saucepan with the remaining hot milk, whisking continuously as you pour.
Cook the mixture over medium-low heat, whisking constantly and thoroughly, until it thickens significantly. Do not stop whisking at any point during this stage. The cream will go through a thin watery phase and then quite suddenly begin to thicken. Continue whisking until the cream is very thick and holds clear lines when the whisk is drawn through it. It should be thick enough that a spoonful dropped back into the pot holds its shape briefly before settling. Remove from heat immediately.
Allow the cream to cool for 10 to 15 minutes, stirring occasionally to prevent a skin forming on the surface. It will thicken further as it cools.