Theme a field widget with twig

Question

Implementing hook_field_widget_single_element_WIDGET_TYPE_form_alter is altering the widget. It makes sense to theme the changes in a twig template. Here's the widget form alter code:

/**
 * Implements hook_field_widget_single_element_WIDGET_TYPE_form_alter().
 */
function my_module_field_widget_single_element_entity_browser_entity_reference_form_alter(array &$element, FormStateInterface $form_state, array $context) {
  $field_name = $context['items']->getName();
  $terms = $element['entity_browser']['#default_value'];
  if (!isset($field_name) || !isset($terms)) {
    return;
  }

  foreach ($terms as $term) {
    $parent = \Drupal::entityTypeManager()
      ->getStorage('taxonomy_term')
      ->loadParents($term->id());
    $parent = reset($parent);
    $term->setName(trim($term->getName()) . ' | ' . trim($parent->getName()));
  }
}

How do I do pass values like $parent->getName() to a twig template in a custom module?