In a small saucepan, stir together the cornstarch, sugar, and salt. In a bowl, beat the egg well with a whisk; it should be frothy. Gradually whisk the milk into the egg and beat it well to combine.
Pour a little bit of the milk mixture into the dry ingredients in the saucepan and whisk until a smooth paste is formed, with no lumps. Add the rest of the milk and whisk until it is well combined.
Set the saucepan over medium heat and cook, stirring occasionally at first then whisking constantly as the milk gets hot, until the pudding thickens and starts to bubble.
Turn the heat lower, and cook for 1 minute, whisking constantly. Remove the saucepan from the heat and stir in the butter and vanilla.
Let the pudding cool for 5 minutes, whisking twice during that time to prevent a skin from forming. Pour the pudding into a bowl. (For an extra silky smooth pudding, you can pass it through a fine meshed sieve into the bowl.) Place a piece of plastic food wrap directly onto the surface of the pudding, making sure the wrap goes right up to the edges of the bowl. This will prevent a skin from forming on the top.
Once the pudding has cooled to lukewarm, place it in the refrigerator and chill it until it is completely cold, 2 to 3 hours (but there's nothing wrong with eating it while it's still warm, either). Stir the pudding before serving.
Serves 4.
Notes
This recipe makes four servings of homemade vanilla pudding - I recommend you double it while you're dirtying the pot and make enough for eight. I betcha it'll disappear in the wink of an eye. Serve the pudding with some fresh berries on top or with a dollop of whipped cream and some crushed roasted pecans. Or have a plop of it on a pie or cake instead of ice cream. It's pretty fantastic any which way.To get the ultimate smooth pudding, pass it through a fine-meshed sieve either before or after it has cooled. That will remove any bits of thickened egg or soft lumps that might be hiding in the pudding. If you're just going to scarf down big bowlfuls of it, there's no need to sieve it, but if you're serving it for fancy company, well, you know . . . .