Deleted thread - tracking who deleted it

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The cs_posts_deleted_archive table should have a record of when and from where the post was deleted, and what its contents were, but not who deleted it. (note that this works via the Audit Deleted Post Content plugin.)

Which version are you using? Note that as of version 12.0, Auditing would provide exactly the recourse you seek. 

We are running version  11.1.5.13464. 

First ensure "Audit Deleted Post Content" plugin is enabled currently, that would indicate it was active when the post was deleted. Then query your database for the table mentioned above:

SELECT * FROM cs_posts_deleted_archive

If needed you can get the UserId of the user who created the post from Administration > Membership > Members > [select user] > Edit.

Just curious, how/where do we access the database to run SQL queries?  If we are hosted is that something we can do on our own or would support have to do it for us?

Yes, if hosted please contact support/open a ticket for assistance, and include this thread link for reference.