Cake, a members-only platform, brings department store perks to online retail.

The by-appointment-only shop specializes in cakes and cookies sheathed with a thin sheet of soy-based paper, which has been run through a laser printer and saturated with a digital image.

There are variations of Epiphany or king cakes from around the world, and similar pastries abound in Portugal and Greece to ring in the New Year. In Spain, Roscón de Reyes — a doughnut-shaped ...