Whisk the egg yolks and sugar together in a large mixing bowl until the yolks lighten in color and the sugar is completely dissolved. Set aside.
Combine the milk, cream, and nutmeg in a medium saucepan over high heat and bring just to a boil, stirring occasionally. Remove from the heat and gradually stir the hot mixture into the egg yolk and sugar mixture. Return everything to the pan and cook until the mixture reaches 160°F on a digital instant-read thermometer.
Remove from the heat, stir in the bourbon, pour into a medium mixing bowl, and refrigerate. Chill the mixture until it reaches a temperature of 40°F, 4 to 6 hours.
Once it's chilled, process the mixture in an ice cream maker according to the manufacture's instructions. Serve as is for soft-serve or freeze in an airtight container for 2 to 4 hours for traditional ice cream.