Could you save a root canal with a broken root?

Probably not: If the root is broken, unlikely, unless the break is at the tip of root. If that is the case a root resection can be done. Oral surgeon, or endodontist.

Fractured roots: Most teeth that have had root canal therapy ; suffer a root fracture are hopeless and have to be extracted. There are surgical procedures depending upon the location of the fracture (crown lengthening; apicoectomy) that might be able to salvage the tooth. The question becomes though, would it be worth it and what is the long term prognosis? Even if possible, probably better to ext ; place implant.

Not likely: Some cracked teeth with broken roots may not be treatable.