Bring milk and first quantity of sugar (5 oz) to a simmer over medium high heat. Whisking constantly to avoid burning.
In a separate, large heat proof bowl, combine the eggs, cornstarch, vanilla extract, and the second quantity of sugar (4 oz), whisk to combine.
Once the milk is simmering, turn off the heat and add 1/4 of your hot milk mixture to the egg mixture VERY slowly while whisking constantly. This is going to temper your egg yolk mixture and avoid curdled scrambled eggs. Whisk until smooth.
Turn the heat back on (medium) and slowly add the mixture to the rest of the milk in your saucepan, whisking constantly. Bring to a simmer.
Once your mixture starts bubbling, continue whisking for 1 minute to ensure the mixture is cooked and properly thickened.
Pour your mixture into a heatproof container. Divide your softened butter into cubes and place them on top of the pastry cream, allowing them to melt. Do not cover.
Allow your pastry cream to cool at room temperature until just warm to the touch. This will take a few hours.
Stir in your butter with a whisk. Don't be worried if it looks curdled at first, this is normal. Just keep whisking until smooth.
Optional: Blend pastry cream with an emersion blender to make it extra smooth.
Cover with plastic wrap so that it's touching the surface of the pastry cream to avoid a skin forming. Chill before using. Can be kept in the refrigerator for 2-3 days.