Shake Shack has never been shy about bold flavors, but when its Korean-inspired menu returns, it doesn’t just flirt with sweet heat—it commits like a non-refundable deposit on a wedding venue.

Shake Shack is serving its iconic burgers with a new twist. The New York-based fast food joint is “bringing the decadent, elevated flavors of French Onion Soup” with a fall-ready menu “that reimagines ...