How does a two-stage supercharger work?
Actually yes.  A two stage supercharger used a compressor with two or more compression stages, often with an inter cooler between them, driven by a transmission with different gear ratios to control the compressor speed.  A schematic of this type of supercharger can be seen here on a Rolls Royce Eagle engine.





On the Supermarine Spitfires powered by the larger Griffon engines the supercharger speed had both a LO and HIGH setting and could either be manually controlled by the pilot or automatically controlled.  The HIGH setting was not recommended at lower altitudes to prevent overboosting during operations there.