As per the updated question, no, you cannot configure Magento to invoke an adminhtml controller from the frontend for security reasons. Put the code in a class, and invoke that class from both the adminthml and frontend controllers separately.
Original answer to the original question regarding invoking one controller from another:
You can invoke a controller just like any other code. Use Dependency Injection to get an instance on your constructor, then invoke a method on that object.
<?php
namespace Vendor\Mod\Foo\Bar;
use Vendor\Mod\Controller\Adminhtml\Custom;
class Foobar {
/** @var Custom */
protected $_custom;
public function __construct(Custom $custom)
{
$this->_custom = $custom;
}
public function foo()
{
$this->_custom->initPage();
}
}