journalctl output for the period in question, taken from 'journalctl -l --since today | grep gnome-shell'.

13:19 is my restart of the polkit service.

The actual authentication did not succeed, aptdaemon did not start and no packages were installed; so I don't know exactly when this window broke - though I think it was after 13:00, and thus much later than the polkit messages at the beginning of the log.  (I certainly didn't have a broken window on my screen for ~2h before filing this bug.)

Apr 15 11:33:29 virgil gnome-shell[4571]: polkitAuthenticationAgent: Received 3 identities that can be used for authentication. Only considering one.
Apr 15 11:33:29 virgil gnome-shell[4571]: pushModal: invocation of begin_modal failed
Apr 15 11:33:29 virgil gnome-shell[4571]: polkitAuthenticationAgent: Failed to show modal dialog. Dismissing authentication request for action-id org.debian.apt.install-or-remove-packages cookie 2-c14dea1098d77930282650c2c7031008-3-811320afb0a7974406f93ce80f7d4594
Apr 15 11:59:56 virgil gnome-shell[4571]: remove_mnemonics: assertion 'label != NULL' failed
Apr 15 11:59:56 virgil gnome-shell[4571]: remove_mnemonics: assertion 'label != NULL' failed
Apr 15 12:20:28 virgil gnome-shell[4571]: pushModal: invocation of begin_modal failed
Apr 15 12:20:28 virgil gnome-shell[4571]: pushModal: invocation of begin_modal failed
Apr 15 12:22:57 virgil gnome-shell[4571]: remove_mnemonics: assertion 'label != NULL' failed
Apr 15 12:22:57 virgil gnome-shell[4571]: remove_mnemonics: assertion 'label != NULL' failed
Apr 15 12:33:28 virgil gnome-shell[4571]: pushModal: invocation of begin_modal failed
Apr 15 12:33:28 virgil gnome-shell[4571]: pushModal: invocation of begin_modal failed
Apr 15 12:53:18 virgil gnome-shell[4571]: JS WARNING: [resource:///org/gnome/gjs/modules/signals.js 128]: Too many arguments to method Clutter.Actor.destroy: expected 0, got 1
Apr 15 12:56:06 virgil dbus-daemon[1964]: [system] Activating via systemd: service name='net.reactivated.Fprint' unit='fprintd.service' requested by ':1.317' (uid=1000 pid=4571 comm="/usr/bin/gnome-shell " label="unconfined")
Apr 15 12:56:12 virgil dbus-daemon[4403]: [session uid=1000 pid=4403] Activating service name='org.gnome.Nautilus' requested by ':1.26' (uid=1000 pid=4571 comm="/usr/bin/gnome-shell " label="unconfined")
Apr 15 12:56:12 virgil dbus-daemon[4403]: [session uid=1000 pid=4403] Activating service name='org.freedesktop.FileManager1' requested by ':1.26' (uid=1000 pid=4571 comm="/usr/bin/gnome-shell " label="unconfined")
Apr 15 12:56:13 virgil gnome-shell[4571]: [AppIndicatorSupport-DEBUG] Registering StatusNotifierItem :1.80/org/ayatana/NotificationItem/software_update_available
Apr 15 12:56:13 virgil gnome-shell[4571]: [AppIndicatorSupport-FATAL] unable to update overlay icon
Apr 15 12:56:13 virgil gnome-shell[4571]: [AppIndicatorSupport-FATAL] unable to update overlay icon
Apr 15 12:56:13 virgil gnome-shell[4571]: Error connecting to Nautilus
Apr 15 12:56:14 virgil gnome-shell[4571]: Couldn’t parse steam.desktop as a desktop file, will treat it as a regular file.
Apr 15 13:00:08 virgil gnome-shell[4571]: JS WARNING: [resource:///org/gnome/shell/ui/workspacesView.js 639]: reference to undefined property 1
Apr 15 13:00:08 virgil gnome-shell[4571]: JS ERROR: TypeError: this._workspacesViews[i] is undefined
Apr 15 13:13:40 virgil gnome-shell[4571]: remove_mnemonics: assertion 'label != NULL' failed
Apr 15 13:13:40 virgil gnome-shell[4571]: remove_mnemonics: assertion 'label != NULL' failed
Apr 15 13:19:17 virgil polkitd(authority=local)[22657]: Registered Authentication Agent for unix-session:2 (system bus name :1.317 [/usr/bin/gnome-shell], object path /org/freedesktop/PolicyKit1/AuthenticationAgent, locale en_US.UTF-8)
Apr 15 13:20:36 virgil gnome-shell[4571]: Could not delete runtime/persistent state file: Error removing file /home/vorlon/.local/share/gnome-shell/notifications: No such file or directory
Apr 15 13:23:50 virgil gnome-shell[4571]: Couldn’t parse steam.desktop as a desktop file, will treat it as a regular file.