Kagi's FastGPT seems to have had this capability for a while. I'd noted a few days ago that a query I'd submitted turned up one of my own two-week-old HN comments amongst its references.

That's not "please find a website" but as a citation supporting its own findings. But either way, either the training set is quite recent or FastGPT conducts searches based on the query and/or results it generates. I suspect the latter.