When players enter Hana House in downtown Brooklyn, New York, they hear mahjong before they see it: a staccato waterfall of plastic clacking together. It’s been the background soundtrack in many Asian households for over two centuries since the four-person tile game was invented in China. But today’s scene is a far cry from an auntie’s living room. 

Enter Green Tile Social Club, New York City’s hot mahjong social club, which celebrated its second anniversary at Hana House this…