If you were a blue-blooded Upper East Side A-lister, glamorous Babe Paley's daughter and on-the-scene Charlie Rose's intermittent inamorata, wouldn't a week in Montserrat seem the proper vacation? For Amanda Burden — who's all that and also Bloomberg's planning commissioner — it's a different story. Turns out Burden was spending her downtime last week in Copenhagen, not so much seeing the sights as trailing new transportation commissioner Janette Sadik-Khan and two of her lieutenants as they studied public squares and waterfronts and bike parking, according to spokespeople at the Planning and Transportation Departments. Grants are paying for the trip for Sadik-Khan and her deputies; Burden, a source tells us, is traveling on her own time and nickel. There was no word on whether Charlie had joined her. —Alec Appelbaum