Add milk into a mixing bowl.
Insert mixer beaters into the mixing bowl, cover the bowl up, and freeze it for a half-hour (or until formation of ice crystals develop).
Afterwards, add the vanilla, cinnamon, and sugar to the mixing bowl.
Beat the mixture until it is fluffy and thick.
Split mix between each mug.
Add coffee to the mugs and sprinkle some nutmeg in.
If desired, the cinnamon sticks can be used to garnish.
Serve immediately.