Chiral surface plasmon polaritons (SPPs) produced by plasmonic nanowires can be used to enhance molecular spectroscopy for biosensing applications. Nevertheless, the switchable stereoselectivity and detection of various analytes are limited by a lack of switchable, chiral SPPs. Using both finite-element method simulations and analytic calculations, we present a graphene-coated chalcogenide (GCC) nanowire that produces mid-infrared, chiral SPPs. The chiral SPPs can be reversibly switched between "on" (transparent) and "off" (opaque) by non-volatile structural state transitions in the dielectric constants of the chalcogenide glass Ge2Sb2Te5. Furthermore, by controlling the Fermi energy of the graphene-coating layer, the nanowire can output either non-chiral or chiral SPPs. A thermal-electric model was built to illustrate the possibility of ultrafast on/off switching of the SPPs at the terminus of the nanowire. Finally, we show that a selective, lateral sorting of sub-10-nm enantiomers can be achieved via the GCC nanowire. Chiral nanoparticles with opposite handedness experience transverse forces that differ in both their sign and magnitude. Our design may pave the way for plasmonic nanowire networks and tunable nanophotonic devices, which require the ultrafast switching of SPPs, and provide a possible approach for a compact, enantiopure synthesis.