Recent studies of cooled oil emulsion droplets uncovered transformations into a host of flattened shapes with straight edges and sharp corners, driven by a partial phase transition of the bulk liquid phase. Here, we explore theoretically the simplest geometric competition between this phase transition and surface tension in planar polygons and recover the observed sequence of shapes and their statistics in qualitative agreement with experiments. Extending the model to capture some of the three-dimensional structure of the droplets, we analyze the evolution of protrusions sprouting from the vertices of the platelets and the topological transition of a puncturing planar polygon.