We report an embarrassingly parallel method for the evaluation of thermodynamic properties over an energy landscape exhibiting broken ergodicity, nested is the likelihood of the observed data D givenbasin-sampling (NBS). We also introduce the No Galilean U-Turn Sampler (NoGUTS), a new sampling scheme based on the No U-Turn Sampler (NUTS) introduced by Hoffman and Gelman (2014) that works with the Galilean Monte Carlo scheme introduced by Betancourt (2012) to aid the efficient generation of new live points. NoGUTS can be thought of as a form of reflective slice sampling with an automatic stopping criterion. We apply this approach to a benchmark atomic cluster of 31 Lennard-Jones atoms, which exhibits a low temperature solid-solid heat capacity peak. The calculated heat capacity is compared with results generated by parallel tempering (PT), basin-sampling parallel tempering (BSPT), and standard nested sampling (NS) simulations. NBS reproduces the full heat capacity curve predicted by PT and BSPT, while the NS calculation with similar computational cost fails to resolve the low-temperature solid-solid phase transition.