Louis Langrée is no stranger to David Geffen Hall. For nearly two decades he’s been its summer tenant as music director of the Mostly Mozart Festival. And the Cincinnati Symphony Orchestra, which he has led since 2013, has even stopped by for a visit.

But it wasn’t until Thursday that Mr. Langrée, 59, made his conducting debut with the hall’s house band, the New York Philharmonic. The orchestra should be thinking about how soon he can return.