What is consular processing and how does it help to claim days left in my 6 year period on H1B?

Attorney Brown is absolutely correct - to claim days left you would need to disclose them on the H-1B petition.

It does not. The processing is when you attend a consulate and get the visa instead of the employer's filing for extension of her stay here. It is applicable only if you fell out of status for short period of time and extension is not possible.

Employer seeks to recapture time, if any, left from the six years allotted to an H-1B employee