The Fluorescence detector Array of Single-pixel Telescopes (FAST) is one of several proposed designs for a next-generation cosmic-ray detector. Such detectors will require enormous collecting areas whilst also needing to remain cost-efficient. To meet these demands, the FAST collaboration has designed a simplified, low-cost fluorescence telescope consisting of only four photomultiplier tubes (PMTs). Since standard air shower reconstruction techniques cannot be used with so few PMTs, FAST utilises an alternative two-step approach. In the first step, a neural network is used to provide a first estimate of the true shower parameters. This estimate is then used as the initial guess in a minimisation procedure where the measured PMT traces are compared to simulated ones, and the best-fit shower parameters are found. A detailed explanation of these steps is given, with the expected performance of FAST prototypes at the Telescope Array experiment acting as a demonstration of the technique.