How to move Drupal 8 with composer, to another server
As already said in the comments composer does not have a way to help with transfering a site.

With a site build with composer I usually zip the composer directory (which includes the drupal directory) and get a database dump.  On the receiving server just copy the files on a similar directory structure and import the database.

For going from plesk to cpanel you probably have to set the document root correctly.  In plesk its usually httpdocs/docroot but in cpanel it should probably be set to public_html/docroot 

Also I don't suppose you will need but it does not hurt to run composer update

NOTE: this is a quick and dirty way of transferring from on host to the other assuming that php version and other parameters stay the same, this is by no means a best practice in transferring a site to a new host.

For best practice you should follow the advice in the comments and see the guide on Use Composer to manage drupal site dependencies and the section for Managing existing sites using Composer might come in handy.