How you complain about a solicitor or barrister will depend on which Australian state or territory you are located.
Typically however, the Law Society/Institute in your State or Territory will be able to provide details on how to complain about a solicitor, and the Bar Association in your State or Territory will be able to provide details on how to complain about a barrister.
Below are links to the State/territory Law Societies.
All solicitors have professional indemnity insurance, which is required prior to them being provided their practising certificate.
The insurers also require solicitors to not only inform them of formal complaints but also potential complaints.
Other avenues first
Typically, the insurer would then assign another solicitor to act on behalf of the insurer to try and resolve the complaint as expeditiously as possible.
In some cases, for various reasons including fear of premium increases, solicitors may not inform the insurer of a potential claim until it becomes formalised.