Usually the cylinder for Schlage doorknobs can be removed by turning the key in the knob and pushing a spring-loaded retainer to pull the knob off. On really old Schlage knobs this retainer wasn’t there, you had to “backdoor” the knob: completely disassemble it from the rear. This is much harder than taking the cylinder out of modern commercial knobs. In fact backdooring a Schlage A series was part of the test for the NWLA certified professional locksmith certification.
I can’t find any identifying information about this knob, I can only say that it is the predecessor to the modern a series. I put this here in case anybody runs into one and wants to know how to take it apart, the owner of this one told me that two locksmiths had told him it couldn’t be rekeyed. It absolutely can. The cylinder can even be replaced. I hope that these pictures help somebody!