By Richard Morgan The Wall Street Journal September 25, 2012 As far back as the race between the Chrysler Building and the Empire State Building, New York has been a city built on one-upmanship. Now that mentality has crept into the cozy demimonde of neighborhood coffee shops. Jack's Stir Brew Coffee opens a new location on Sixth Avenue at Downing Street next month, reviving bragging rights over a plaque that hangs in its original West 10th Street location: a patent by owner Jack Mazzola Jr. Unable to find a machine to produce coffee precisely the way he wanted, he created one that is true to his shop's name—it stirs coffee as it is brewing, which Mr. Mazzola says distributes flavor more evenly and robustly. And when Fair Folks Cafe opens Thursday on West Houston Street, it will offer a new kind of bragging rights: For a monthly fee of $25, patrons can become members of the coffee shop, getting free drinks daily, as well as discounts on art and design produc...