It's a popular winter speciality of Milan, invented in the Caffè dei Virtuosi by waiter Domenico Barbaja (or Barbaia) in the early 1800s: espresso, cocoa and milk, with sugar and sometimes cream on top, considered a forerunner of the cappuccino.
Also spelled barbajada.
Domenico Barbaja, who had no small gifts for self promotion and seizing every opportunity that passed his way, went on to fame and fortune. First he opened a chain of coffee shops across Milan selling his concoction, became an arms dealer during the Napoleonic wars.
After the wars, he switched to running casinos, earning enough money to purchase the Teatro San Carlo in Naples as well as two other opera houses in Vienna, and eventually managing La Scala in Milan as well, commissioning operas from Gaetano Donizetti, Vincenzo Bellini Carl Maria von Weber and especially Gioacchino Rossini.
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