We study two-dimensional (2D) anisotropic droplets of non-interacting electrons in the lowest Landau level, confined by trapping potentials whose level curves have an arbitrary shape at large distances. Using semiclassical methods, we show that energy eigenstates are localized on equipotentials of the trap, with angle-dependent local widths and heights. We exploit this one-particle insight to deduce explicit formulas for many-body observables in the thermodynamic limit. For instance, the droplet's density falls off at the boundary with an angle-dependent width inherited from that of the underlying wave functions, while the many-body current is localized on the edge, to which it is tangent. Correlations along the edge are long-ranged, in accordance with the system's low-energy edge modes which are described by a free chiral conformal field theory in terms of the angle variable of the trapping potential. These results are likely to be observable in solid-state systems or quantum simulators of 2D electron gases with a high degree of control on the confining potential.