A thin water film evaporating from a cleaved mica substrate undergoes a first-order phase transition between two values of film thickness. During evaporation, the interface between the two phases develops a fingering instability similar to that observed in the Saffman-Taylor problem. The dynamics of the droplet interface is dictated by an infinite number of conserved quantities: all harmonic moments decay exponentially at the same rate. A typical scenario is the nucleation of a dry patch within the droplet domain. We shall construct solutions of this problem and analyze the toplogical transition occuring when the boundary of the dry patch meets the outer boundary. We shall show a duality between Laplacian growth and evaporation, and utilize it to explain the behaviour near the transition. To this end we construct a family of problems for which evaporation and Laplacian growth are limiting cases and show that a necessary condition for a smooth topological transition, in this family, is that all boundaries share the same pressure.