In a saucepan bring the milk and cream to a boil and add 3T sugar. While this is heating, thoroughly mix egg, 1T sugar, cornstarch.
Once the milk has boiled, remove it from the heat and slowly stream it into the egg-cornstarch-sugar mixture while stirring. You're trying not to cook the eggs. Once it's mixed, put it back in the saucepan and bring it back to a boil while stirring constantly.
When it thickens, around when it gets back to the boil, take it off the heat and stir in the butter and vanilla. Put into a pyrex dish, cover the top with cling-wrap that's right on the surface (to avoid forming a film) and put it in the fridge to chill. If you're in a hurry you can use a baking sheet.