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The ability to (re)establish basic community infrastructure and governmental functions, such as medical and communication systems, after the occurrence of a natural disaster rests on a continuous supply of electricity. Traditional energy-generation systems consisting of power plants, transmission lines, and distribution feeders are becoming more vulnerable, given the increasing magnitude and frequency of climate-related natural disasters. We investigate the role that fuel cells, along with other distributed energy resources, play in post-disaster recovery efforts. We present a mixed-integer, non-linear optimization model that takes load and power-technology data as inputs and determines a cost-minimizing design and dispatch strategy while considering operational constraints. The model fails to achieve gaps of less than 15%, on average, after two hours for realistic instances encompassing five technologies and a year-long time horizon at hourly fidelity. Therefore, we devise a multi-phase methodology to expedite solutions, resulting in run times to obtain the best solution in fewer than two minutes; after two hours, we provide proof of near-optimality, i.e., gaps averaging 5%. Solutions obtained from this methodology yield, on average, an 8% decrease in objective function value and utilize fuel cells three times more often than solutions obtained with a straight-forward implementation employing a commercial solver.