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1.

<input type="submit" class="btn btn-danger btn-sm mb-3" name="my_submit_form" value="Send data"/>

2.

class AdminOrderManagerController extends ModuleAdminController
....
....
public function initContent()
{
    $orderId = ''; /* ????????????????????????? */
    $this->context->smarty->assign(
        array(
            'order_ID' => $orderId,
            '_form_action' => $this->context->link->getAdminLink('AdminOrderManager'),
        )
    );

    $this->content = $this->module->fetch('module:mymodule/views/templates/front/added.tpl').$this->postProcess();
    parent::initContent();
}

public function postProcess() {
    if (Tools::isSubmit('my_submit_form')) {
        if (Tools::getValue('order_ID')) {
            // your function
        }
    }
}

1.

<input type="submit" class="btn btn-danger btn-sm mb-3" name="my_submit_form" value="Send data"/>

2.

class AdminOrderManagerController extends ModuleAdminController
....
....
public function initContent()
{
    $orderId = ''; /* ????????????????????????? */
    $this->context->smarty->assign(
        array(
            'order_ID' => $orderId,
            '_form_action' => $this->context->link->getAdminLink('AdminOrderManager'),
        )
    );

	$this->content = $this->module->fetch('module:mymodule/views/templates/front/added.tpl').$this->postProcess();
    parent::initContent();
}

public function postProcess() {
    if (Tools::isSubmit('my_submit_form')) {
        if (Tools::getValue('order_ID')) {
            // your function
        }
    }
}