Moreover, in households from medieval times all the way up through the Victorian era, servants were never supposed to be seen by residents in most cases, unless they were direct personal servants. Messes were, in effect, supposed to seem like they magically cleaned themselves, with entire staircases, hallways, etc. all present, almost a shadow household to keep the servants out of sight. Many of them would do the majority of their work only after the "masters" had gone to bed.
So the idea that you have a cook--or some other servant--awake at 2:30 doing something generally in that area? Not only possible, extremely likely because of the social mores of medieval, renaissance, and early-modern Europe.