Next, make the pudding.
In a medium bowl, mix together ¼ cup of the milk and the egg yolks. Set aside.
In a medium saucepan, add the remaining milk, sugar, cornstarch, and salt over medium heat.
Stirring often, bring to a light simmer.
Add ½ cup of the warm milk mixture to the egg yolks, whisking to combine.
Add another ½ cup of the warm milk to the egg mixture another two times (three times total). This tempers the egg yolks so they don’t curdle.
Add this mixture then to the remaining milk in the saucepan.
Lower the heat to medium-low and continue heating another 4-6 minutes, stirring with a spatula, until it is thick enough that it covers the back of the spatula.
Remove from heat and stir in butter and vanilla, stirring until butter is melted.
Cover with plastic wrap, pressing it directly on the surface of the pudding to avoid it developing a skin, and let cool completely.