Recent years have seen the development of several new computational techniques for solving stochastic optimal control problems by parametrizing the control using neural networks and optimizing the objective empirically on sample paths. In the setting of optimal stopping, we instead parametrize the free boundary, from which the control can be reconstructed. This provides better interpretability of the solution as well as topological guarantees for its structure. The method is both reliable and performant, and it suggests similar ideas could be implemented in other free boundary contexts.