Whisk 1 egg, 1 ounce Irish cream liqueur and 1/4 ounce Irish whiskey in a bowl until smooth. Bring 2 cups whole milk to a boil in a saucepan (do not use a slow cooker-it will overcook the eggs). Pour over the egg mixture, stirring to mix well. Pour into mugs. Sprinkle with nutmeg and garnish with cinnamon sticks.
If you don't have a slow cooker, keep these drinks warm in a saucepan on the stovetop over low heat.