Pour the milk into a small saucepan; add the white chocolate chips and 1/4 tsp of vanilla extract.
Heat over medium-low heat, stirring often, just until the white chocolate is melted (do not allow to boil). The milk will get hotter while it's steaming later.
Fill your espresso maker filter holder with the ground espresso and pack it firmly in place.
Place the filter holder into the machine and lock it in place.
Fill the frothing pitcher with water and pour it into the machine and replace the lid.
Turn the machine to the espresso setting to begin brewing espresso.
Once the brewing begins, and the espresso has started dripping into the metal carafe, flip the knob to the steaming setting.
Steam the white chocolate milk, gently moving the coffee mug slightly up and down (to heat the milk through), as well as letting the tip of the steam rise just above the milk to get a nice froth. Finish the steaming process once you have a nice froth and the milk is hot.
Turn the knob back to the espresso setting and let the remaining espresso drip into the carafe.
Flip back to the steaming setting to release all of the pressure from the steamer.
Carefully pour the espresso into the steamed & frothy milk, to taste. Enjoy.