Put 1 cup of the milk and the half and half in a saucepan. Run the tip of a sharp knife along the length of the vanilla bean, split it open and use the side of the knife to scrape the seeds into the pan. Add the bean along too. Heat just until it comes to a simmer over medium heat.
While that is heating, whisk together the sugar, cornstarch, and salt in a medium heatproof bowl.
Whisk in the remaining 2/3 cup whole milk, and then the eggs. Once the milk has just come to a simmer, very gradually drizzle it into the cornstarch mixture in the bowl, whisking the whole time.
Return the mixture back into the saucepan, stirring constantly with a rubber spatula or whisk.
Once it comes to a simmer, cook it for one minute longer, stirring constantly. Stir in the vanilla extract and bourbon and divide the pudding among 5 or 6 dishes.
I like to eat this pudding warm, but you can chill it in the fridge to set it up. Whether you eat it warm or cold, you'll get the nice follow up warmth of the bourbon in every bite.