No input file specified - [proxy_fcgi:error] Got error 'Unable to open primary script: [PATH]/html/index.php (No such file or directory)\n'
It turned out that the problem was from customizing the domain's document root in the userdata files and then the PHP-FPM configuration had not been rebuilt and the service not restarted.  The following resolved the issue:

/scripts/php_fpm_config --rebuild
/scripts/restartsrv_apache_php_fpm


It's interesting since there are a broad variety of issue that seem to cause this No input file specified issue, but they all mostly revolve around Apache config issues.