  We experimentally and theoretically investigate the behavior of Leidenfrost
droplets inserted in a Hele-Shaw cell. As a result of the confinement from the
two surfaces, the droplet has the shape of a flattened disc and is thermally
isolated from the surface by the two evaporating vapor layers. An analysis of
the evaporation rate using simple scaling arguments is in agreement with the
experimental results. Using the lubrication approximation we numerically
determine the shape of the droplets as a function of its radius. We furthermore
find that the droplet width tends to zero at its center when the radius reaches
a critical value. This prediction is corroborated experimentally by the direct
observation of the sudden transition from a flattened disc into an expending
torus. Below this critical size, the droplets are also displaying capillary
azimuthal oscillating modes reminiscent of a hydrodynamic instability.
