Ella Fitzgerald, Chet Baker, even Amy Winehouse, these are some of the few jazz singers who are household names—and none of them debuted this decade. With the dominance of pop, house and hip hop, it looked like the genre of jazz was going to be relegated to yesteryear, reserved only for elevators or Italian restaurants. Until Laufey.

In addition to being one of the most exciting jazz songstresses of her generation, Laufey is a generational dream girl. Between her velvety...