Set a fine mesh strainer over a medium sized bowl and set the bowl over a large container of ice water.
In a medium saucepan over medium heat, heat the milk, cream, and 1/2 cup sugar, stirring occasionally until steam appears and the liquid is hot (175 degrees) about 5 minutes.
When the milk mixture is cooking, in a separate bowl, whisk the yolks and 1/4 cup of sugar together until smooth. Slowly whisk about 1 cup of the hot milk mixture into the yolks to temper them. Then slowly whisk the tempered yolk mixture back into the remaining hot milk mixture. Continue to cook the custard mixture over medium heat until it is very hot but not simmering (180-185 degrees).
Pour the custard mixture into the strainer bowl set in the ice bath and let cool, stirring occasionally, for about 10 minutes. Remove the custard mixture from the ice bath and add nutmeg and bourbon. Cover tightly with plastic wrap and refrigerate until very cold (40 degrees), about 3 hours.
Remove from refrigerator and add the vanilla extract; stir well. Pour the custard into the ice cream machine canister and churn, following the manufacturer’s instructions, until the mixture resembles soft-serve ice cream, about 20 minutes. Transfer the ice cream to an airtight container, press plastic wrap flush against the surface, cover the container, and freeze the ice cream until firm, at least 2 hours.