This WordPress instance was recently migrated from shared hosting to a VPS, and everything appears to be working properly now and is very fast, with two funny exceptions:
1. When I click on the "My Sites" dropdown at the top of the page and then select one of the subsites, the meat of the page takes almost exactly 60 seconds to load. This appears to happen with only one of the subsites.
2. When I am in that same subsite and click the "upload an image" icon for a post (new or existing), the popup containing the upload dialog takes almost exactly 60 seconds to load.
I don't think this is a coincidence. This is not repeatable every time, perhaps because of something that's being cached; however, when it does happen, I can see a mysql sleep process take almost exactly 60 seconds too, terminating when the process in either exception #1 or #2 finishes loading.
I have also separately migrated this WordPress instance to another shared hosting account at a third hosting provider, and I don't see this problem.
I have tried disabling all plugins, changing all wait/timeout settings I could find in PHP and MySQL, especially any 60-second ones, and that did nothing to fix the speed problem for these functions. I have attempted to place debug scaffolding in the code to see where there slowdown occurs; but since it loads quickly on subsequent clicks, it's difficult to pinpoint.
Any ideas on what this could be?