When GaAs is heated in vacuum, it decomposes into Ga and As as it evaporates. Real-time in situ surface electron microscopy reveals striking bursts of "daughter" droplet nucleation and growth when coalescence of large "parent" droplets exposes nonplanar surface regions. We analyze the behavior, predicting a morphology-dependent congruent evaporation temperature. Based on this we propose a new approach for the self-assembly and positioning of quantum structures via droplet epitaxy, which we demonstrate at the proof-of-concept level. © 2010 The American Physical Society.