In a medium-sized, heavy-duty metal saucepan, combine the sugar and cornstarch. Add the half and half to the pan and place it over medium heat on the stove. Stir gently with a whisk as the mixture heats up, continuing until it thickens and comes to a boil. Once it boils, whisk slowly and let it boil for an additional two minutes before removing the pan from the heat.

Take one cup of the thickened mixture out of the pan. Slowly pour it into the egg yolks while stirring gently to combine. Once mixed, gradually whisk the egg yolk mixture back into the pan, adding it slowly to avoid curdling.

Return the pan to the stove over medium heat. Bring the mixture back to a gentle boil, then reduce the heat and continue to cook, whisking slowly for another two minutes.

Remove the pan from the heat and stir in the butter, salt, and vanilla extract. Pour the pudding into dessert dishes and cover the surface with plastic wrap, ensuring it touches the pudding to prevent a skin from forming as it cools. Allow the pudding to cool on the counter for 45 minutes before refrigerating for up to one day.