Daniel Lee, formerly the creative director of Bottega Veneta, has ambitious aspirations for Burberry.

In the penthouse apartment at Claridge's, Daniel Lee wears an army green pullover, black athletic leggings, and white Nikes. It is the middle of November, and through the French doors behind him he can see a panorama of a rainy London, with Big Ben in the background. Lee's throat is a little sore because he spent the previous evening at Chiltern Firehouse, where he ran across local fashion writers and at least one of his former teacher....