Had a nursing home with a walk-in like that. The thermostat was in series with the coil of the compressor contactor. When satisfied it would break the line to the coil, compressor would shut off. However there is no pumpdown with this method, so the compressor can try and start against a full load.
This is a good cause of "flooded start" which can damage the compressor. The late Glen Henigan from Copeland did an excellent video about this.
Next time back, explain this to the customer as this is a Compressor Killer.