Do I have any special extra work I need to do, afterward, if I have to restart the Puppet Master service?
Specifically, does this affect SSL certificates?
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Puppet masters can be restarted at will (that is if you don't have any agents trying to run puppet). Nothing bad will happen with any certificates or configuration or anything else.
You will have a problem if you delete the machine with the puppet master and don't have any backups. In this case, after you create your new puppet master, you will have to regenerate the certificates for all machines. (this is simplified, I'm presuming here that you only have 1 puppet master that it's also a puppet ca server)