I'm Ivan, a Product Manager in Jira Software, and I wanted to update everyone that my team is currently working on the ability to manually clear issues in the DONE column on an ad-hoc basis. Please be patient as we just wrapped up our discovery research and design spike, and will be undergoing development work in the next few sprints. The other relevant ticket is here: https://jira.atlassian.com/browse/JSWCLOUD-17582
With regards to finding issues that have been cleared from your DONE column, you will be able to do that through the new ISSUE NAVIGATOR in next-gen as stated here https://community.atlassian.com/t5/Next-gen-articles/Introducing-issue-navigator-in-next-gen-projects/ba-p/1339049, which will ship before manual board clearing.
Regarding bringing already-cleared issues back to the board (which we don't encourage as it might be better to create a new issue instead reflecting the latest thinking about that work), you can simply change the status of the issue.
NOTE: the manual clearing option will only be available in the DONE column on the board view, and we will sync that with the board container in the backlog view. This is not a mechanism to filter out completed issues in the backlog list in the backlog view, as the backlog list should be a list containing TO-DO items, and completed work should not appear there.
Let me know if you got any further questions.
Product Manager, Jira Software