Clean Firefox favicon cache?

I'm not sure if this would be possible with CCleaner or even regarded as something that CCleaner should do if it could, but I would love it if it was capable of cleaning the Firefox favicon cache.

I hate it when a site updates its favicon, but the old one remains on the bookmark toolbar.

I've had to install an add-on called SQLite Manager to achieve it, but doing so is a bit of a chore.

CheckPlaces is another Firefox add-on which can check and update Firefox's stored favicons.