Burnham and Allegro both have a historic feeling, but for different reasons.

Burnham used to be offices, so the hallways have the feel of an old office building, complete with transoms above the doors, etc. If you're in the right mood, you can imagine you're in a film noir going to visit a private eye.

Allego, OTOH, has always been a hotel. Formerly the Bismarck Hotel. It dates from 1926 and has lots of history in its own right.