Place the drive in an external enclosure. I have used ‘recovermyfiles’ software before to recover pictures from a reformatted computer (thankfully they chose the quick format option), but I've never seen it actually recover files from a system that was formatted in a more systematic manner.
As far as I understand it the only difference between quick format and full format is that full format performs error checking. Nothing else, or minimal else is written to the drive. Pls correct if you know otherwise
Good advice to remove the drive. The data is probably still there, at least until it gets overwritten. The trick is being able to read it. Some of the applications already posted will work under the right circumstances.
The quick formats don’t overwrite the data so it will be there until overwritten. If they did a full format then a forensics lab is about your only option.