Most efficient way to use classes to create admin pages using Settings API

The example in Settings API documentation uses admin_init and admin_menu for registering custom settings and admin pages. https://developer.wordpress.org/plugins/settings/custom-settings-page/

There’s also an old Codex entry on actions typically run on front and back end requests, which can help pick a suitable action hook. https://codex.wordpress.org/Plugin_API/Action_Reference

I also highly recommend the excellent old Q&A about instantiating classes in WP. Best way to initiate a class in a WP plugin?