To get a brief reprieve from the pressures of working in the White House, Kristina Schake, a former aide to the first lady, Michelle Obama, took a class about her favorite painter, Michelangelo Merisi da Caravaggio. She noticed that the Italian painter often showed Christ with bare feet, portraying his subject as a common man. It was a lesson that informed Ms. Schake’s job in the East Wing when, as Mrs. Obama’s communications chief, she encouraged the first lady to take an undercover shopping trip to a Target in suburban Alexandria, Va., to showcase her dance moves on “Late Night...