The large-scale geometry of discrete models of random curved surfaces, like the uniform triangulation of the sphere, has been studied intensively. In this talk I will focus instead on random flat surfaces with the topology of the disk, in which all degrees of freedom are located at the boundary of the disk. Addressing a recent question of Frank Ferrari in the physics context of JT gravity, I will present a pair of discrete models of flat disks with controlled boundary length whose enumeration can be established via bijective means, involving planar random walks in one case and labeled planar maps in the other. These observations hint at the existence of a universal continuous flat random disk.