How to print the output of a controller

This is old, but this is how I did it:

    $html = Drupal::service('renderer')->renderRoot($array);
    /** @var PrintBuilder $print_builder*/
    $print_builder = Drupal::service('entity_print.print_builder');
    /** @var RendererInterface $renderer*/
    $renderer = Drupal::service('entity_print.renderer_factory')->create([$entity]);
    $print_engine = Drupal::service('plugin.manager.entity_print.print_engine')->createInstance('dompdf');
    $print_engine->addPage($html);
    $content = $renderer->render([$entity]);
    $render = [
      '#theme' => 'your_template',
      '#title' => $this->t('View @type', ['@type' => $print_engine->getExportType()->label()]),
      '#content' => $content,
      '#attached' => ['your_module/library'],
    ];

    $print_engine->addPage($renderer->generateHtml([$entity], $render, FALSE, TRUE));

    return (new StreamedResponse(function () use ($print_builder, $entity, $print_engine) {
      // The printed document is sent straight to the browser.
      $print_builder->deliverPrintable([$event_notes], $print_engine, FALSE, TRUE);
    }))->send();

As far as I understood, the module requires an entity to work, you can't make it work only with a render array. So I did setup in a random entity a "PDF" view mode (which I knew I wouldn't need to export to PDF later), created the template twig. That template Twig shows nothing, it's just there to make the genereateHtml() method work.

$array is the result of

return [
      '#theme' => 'your_template',
      '#var1' => $var1,
      '#var2' => $var2,
      '#var3' => $var3
    ];

Hope it helps somebody else.