WITH SALES PROJECTED to hit $12.7 billion by 2022 (according to market research provider Euromonitor International) and a vast global social-media footprint, the South Korean beauty industry is serious business—and seriously specific. Korean sheet masks in particular, boasting targeted treatments for every skin concern from head to toe (yes, foot masks exist), have become a slavishly coveted home treatment for the Instagram era, with stars from Bella Hadid to Jimmy Fallon posting ghoulish masked selfies. Barely a blip on the radar of even hard-core American beauty enthusiasts five years ago, these charming envelopes...