Miss Hardbroom appears at the beginning of the first episode.  She is the figure that is conjoured up when Milly looks into the All-Seeing Eye - she is there to warn her about her "mad-cap plans."  She knows Milly's penchant for rushing into things, and tries to warn her, but Milly has matured and now knows that she will be the only one to blame if things go wrong.  So Miss Hardbroom leaves with a smile and wishes her good luck.

 

Clcik here for a more detailed Character Bio of HB:  http://www.miss-cats-worst-witch-academy.de/History__-_Who_is_who_/The_Staff/the_staff.html#misshardbroom