When an operation on the server is slow, please take a set of the server thread dumps (10+) spread over the time of the slowness. TeamCity automatically saves thread dumps on super slow operations. so there might already be some.
It is recommended to send us an archive of the entire content of server's <TeamCity Home>/logs/threadDumps-<date> directories for all the recent dates.
It is recommended that you take a thread dump of the TeamCity server from the Web UI (if the hanging is local and you can still open the TeamCity Administration pages): go to the Administration | Server Administration | Diagnostics page and click the Save Thread Dump button to save a dump under the
<TeamCity Home>/logs/threadDumps-<date> directory (where you can later download the files from "Server Logs").
If the server is fully started but web UI is not responsive, try the direct URL using the actual URL of your TeamCity server.