Stroll through London’s famed Burlington Arcade in the heart of Mayfair, an enclosed array of exclusive shops that has catered to well-dressed gentlemen for over 200 years, and one’s eye cannot help but be drawn to the extraordinary display of watches at the far end—thousands of rare Rolexes, each seemingly more valuable than the next. This is The Vintage Watch Company, which celebrates its 25th anniversary this year, a mecca for Rolex collectors, who include some of the most stylish men on the planet.