Set butter out at room temperature.
Whisk egg and egg yolks together until incorporated.
In a small pot, heat up the Baileys liqueur and sugar on medium heat until the mixture starts to steam and you can see a few bubbles rise up from the sides of the pot. DO NOT allow the mixture to come to a full boil.
SLOWLY pour 1/3 of the liquid mixture into the egg yolks and egg mixture while whisking vigorously to prevent cooking the eggs.
Then take this egg mixture and pour it back into the pot with the liquid mixture. do it SLOWLY while whisking vigorously.
Put it back onto the stove on medium heat. Stir to prevent eggs from curdling. Cook the mixture until it reaches 160 degrees. Take off the heat immediately.
Pour mixture through a strainer into a mixing bowl. Use a spoon to push it through.
In a stand mixer, using a balloon whisk, whisk the mixture on medium speed until it is cool to the touch. It will look lighter in color and the volume will have increased but it will still be runny.
Change to the paddle attachment. On slow speed, add butter a little at a time. Once incorporated, beat on medium speed until fluffy.