Ah, you have a non-extbase extension whose content is cleanly structured into Classes, Configuration, Resources, etc. But then there are these pesky modX/index.php files for your backend modules. That's not nice, but there is help: PSR-7. This tutorial shows how to setup your backend modules according to PSR-7.

You probably set up your backend modules this way in your ext_tables.php:

$extPath = \TYPO3\CMS\Core\Utility\ExtensionManagementUtility::extPath($_EXTKEY);

\TYPO3\CMS\Core\Utility\ExtensionManagementUtility::addModule(
    'web', 'txmoduleM1', '', $extPath . 'mod1/', [
        'access' => 'group,user',
        'name' => 'web_txmoduleM1',
        'labels' => [
            'tabs_images' => [
                'tab' => 'moduleicon.gif',
            ],
            'll_ref' => 'LLL:EXT:huselpusel/mod1/locallang.xlf',
        ]
    ]
);

The hard-coded extension path looks not so nice and even triggers an entry in the deprecation log:

To achieve this, the first step is creating a new controller, let's call it Classes/Controller/BackendModuleController.php:

<?php
namespace Foobar\Huselpusel\Controller;

use TYPO3\CMS\Backend\Module\BaseScriptClass;

class BackendModuleController extends BaseScriptClass
{
}

The next step is inserting a public method that is called for dispatching the module. Let's call it mainAction:

public function mainAction(
    \Psr\Http\Message\ServerRequestInterface $request,
    \Psr\Http\Message\ResponseInterface $response
) {
    // Logic goes here...
}

mainAction is the entry point to the backend module which takes the ServerRequestInterface and ResponseInterface of PSR-7 as parameters. This method handles the logic of your backend module. There is one important thing that behaves completely different: content is not echoed anymore (most likely by the method printContent()), you must use the ResponseInterface object.

You may get your GET and POST parameters with the request object:

$get = $request->getQueryParams();
$post = $request->getParsedBody();

To return the content, you may write into response's body and return the object:

$response->getBody()->write($this->doStuff());
return $response;

You may also define a HTTP status code or set the content type:

if ($FAIL) {
    $response = $response->withStatus(500);
} else {
    $response->getBody()->write($this->doStuff());
    $response = $response->withHeader('Content-Type', 'text/html; charset=utf-8');
}

The content type for backend modules is by default text/html, changing this is more important for AJAX calls based on PSR-7, but this will be explained in another blog post (hint hint).

So far, so good. Let's get back ext_tables.php where the module is registered. The fourth parameter of ExtensionManagementUtility::addModule() requires the path to the module. Drop this, it must be an empty string now. As a replacement, adjust the configuration array by the new key routeTarget:

[
    'routeTarget' => \Foobar\Huselpusel\Controller\BackendModuleController::class . '::mainAction',
    'access' => 'group,user',
    'name' => 'web_txmoduleM1',
    //...
]

After clearing the caches of TYPO3, the backend module is now called by the PSR-7 way and there should be no new entry in the deprecation log. You may copy the whole logic of your former modX/index.php which one exception:

$SOBE = GeneralUtility::makeInstance('tx_huselpusel_module1');
$SOBE->main();
$SOBE->printContent();

Drop that code, you don't need it anymore as the module is already dispatched with your mainAction.

You may move the rest of mod1 into their appropriate location:

moduleicon.gif => Resources/Public/Images/Icons/BackendModuleController/ModuleIcon.gif
locallang.xlf => Resources/Private/Language/locallang_mod.xlf

The whole registration should now look like this:

\TYPO3\CMS\Core\Utility\ExtensionManagementUtility::addModule(
    'web', 'txmoduleM1', '', '', [
        'routeTarget' => \Foobar\Huselpusel\Controller\BackendModuleController::class . '::mainAction',
        'access' => 'group,user',
        'name' => 'web_txmoduleM1',
        'labels' => [
            'tabs_images' => [
                'tab' => 'EXT:huselpusel/Resources/Public/Images/Icons/BackendModuleController/ModuleIcon.gif',
            ],
            'll_ref' => 'LLL:EXT:huselpusel/Resources/Private/Language/locallang_mod.xlf',
        ]
    ]
);

For TYPO3 v8, the registration slightly differs:

\TYPO3\CMS\Core\Utility\ExtensionManagementUtility::addModule(
    'web', 'txmoduleM1', '', '', [
        'routeTarget' => \Foobar\Huselpusel\Controller\BackendModuleController::class . '::mainAction',
        'access' => 'group,user',
        'name' => 'web_txmoduleM1',
        'icon' => 'EXT:huselpusel/Resources/Public/Images/Icons/BackendModuleController/ModuleIcon.gif',
        'labels' => 'LLL:EXT:huselpusel/Resources/Private/Language/locallang_mod.xlf',
    ]
);

After that, you can drop the module folder as it's not required anymore.