I'd also like to use the local one hosted on my own box. Unfortunately, this doesn't seem straight forward because the asset is delivered as a base64 encoded blob, rather than just being a linked thing. Is there a direct link stashed away somewhere?
(Am using nginx + php-fpm)
If I'm understanding the question correctly, you want to have the Eye logo in posts made by a bot in [something that's just like] Slack.
Looks like http://your-phabricator.host/rsrc/image/logo/light-eye.png works to deliver this file in most cases, but this is a more-or-less a hack that might stop working some day. Otherwise, you can just copy the file somewhere, or use the favicon.