Brick Lane Curry House, with an $18 lamb vindaloo and a lack of Christmas lights, was always the sedate, upscale neighbor among the South Asian restaurants on East Sixth Street in Manhattan.

Now, a hellishly hot curry called phaal is drawing a rowdier crowd. On a busy weekend, the dining room teems with young men high-fiving each other, gourmands shooting YouTube videos and ruddy-cheeked diners posing for photos.

Phaal,...