Could you save a root canal with a broken root? is there a procedure for that?

Generally not: If the root is fractured either vertically or horizontally, it is difficult to get a seal that will contain the infection. Consider extraction and evaluation for a bridge or implant

Fractured Root: In most cases no. The fracture would have to be horizontal and near the crest of the bone ridge. Even then it would require another surgical procedure and a poor long term prognosis. In these type of situations, the more conservative treatment and one that generally offers a good long term prognosis is an implant retained crown. There may be even additional replacement options. Get a consult.