To make vanilla pudding, put aside 1/2 cup of milk. Meanwhile, in a large saucepan, combine the remaining milk, cream, salt, and sugar and let it simmer on medium heat. Keep stirring the mixture till the sugar dissolves. Remove the saucepan from the stove and allow it to cool for some time. 

In the meantime, take cornstarch in a bowl, add in the milk which was kept aside and whisk it till you get a smooth consistency. Thereafter, break the eggs and separate the yolks. Add the yolks to the smooth mixture. Continue to whisk till the mixture is further smoothened. 

Now, take the simmering milk from the stove and add it slowly to the yolk mixture, while whisking it to ensure the eggs do not scramble. Once done, transfer the mixture into a saucepan and let it simmer over medium heat. Ensure that the mixture is continuously whisked during this process. You'll notice that it will begin to thicken. Take a thermometer and note the temperature. It should register 180-degree Fahrenheit. Once done, remove the mixture from the stove and add in the butter and vanilla extract. Whisk it till it becomes smooth and shiny. Use a fine-mesh strainer to strain the pudding into a clean bowl/cup. 

Once it is prepared, allow it to cool and then refrigerate it for 3-4 hours or overnight.  Make sure that you whisk the pudding properly before serving so that it smoothens. If there is leftover pudding, it can be stored in an air-tight container for 5-7 days.