I can't reproduce anything based on the steps you've provided here. Here's what I did on this install:

Click a Project tag: https://secure.phabricator.com/tag/diffusion/
Click "Manage Project"
Click "Edit Menu"

The menu page appears for me. What steps am I missing? What Phabricator version specifically are you on? Did you restart Apache or PHP-FPM?

I'm at version a9e2e6c5aab94c8f4c57797ac63bfd88955a630d and php-fpm restarted after upgrading.
I don't have any editing permission on this install obviously, but the reproduce steps are the same. I see a 404 Not Found error on page after clicking "Edit Menu" rather than anything else.

I also can't reproduce this here or locally. I tried several different projects, and tried users with and without permission to edit the menu.

Can you make a movie or give screenshots of where you are clicking?

Here is a screen record.

I still can't reproduce the error. And it happens on this install for you as well?

I opened policy up on https://secure.phabricator.com/project/view/1411/ for testing.

It works well on this site... There must be something strange on my own install...

os/php ?

Hi, I think I find the cause.

In this file https://secure.phabricator.com/diffusion/P/browse/master/src/applications/search/engine/PhabricatorProfilePanelEngine.php;6349741760a84de46f6a53d57fadef3b044b15ee$68.

After enabling phabricator.show-prototypes, I can see the configuration page. But according to 2016 Week 4 (Late January), the feature should be in stable, isn't it?

Thanks for digging into this more @clippit.

Thanks for your great work too :)