How to check menu item class in Walker_Nav_Menu

I have a custom two level menu in WordPress. There is an upper level and when you hover over the items, a submenu appears. Two menu items in the submenu have a button that is not in the other submenus. These two paragraphs have a "browse all" class. I need to check this class in Walker_Nav_Menu and add a custom button to the submenu. How can I check for class "browse all"?
In my code I'm creating a wrapper for ul.sub-menu. I need to check if there is a "browse all" class in the element in order to add a button to this wrapper. Such a button will only be in items with the "browse all" class.

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<!-- language: lang-js -->

class My_Walker extends Walker_Nav_Menu {
  function start_lvl( & $output, $depth = 0, $args = array()) {
    $indent = str_repeat("\t", $depth);
    if ($depth == 0) {
      $output. = "\n$indent<div class='sub-menu__depth-1'><ul class='sub-menu sub-menu__main'>\n";
    } else {
      $output. = "\n$indent<ul class='sub-menu'>\n";
    }
  }

  function end_lvl( & $output, $depth = 0, $args = array()) {
    $indent = str_repeat("\t", $depth);
    if ($depth == 0) {
      $output. = "$indent</ul> <
        /div>\n";
    } else {
      $output. = "$indent</ul>\n";
    }
  }
}

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Below is an example of how it should be:

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<!-- language: lang-js -->

class My_Walker extends Walker_Nav_Menu
{
	function start_lvl(&$output, $depth = 0, $args = array())
	{
		$indent = str_repeat("\t", $depth);
		if ($depth == 0) {
			$output .= "\n$indent<div class='sub-menu__depth-1'><ul class='sub-menu sub-menu__main'>\n";
		} else {
			$output .= "\n$indent<ul class='sub-menu'>\n";
		}
	}
	function end_lvl(&$output, $depth = 0, $args = array())
	{
		$indent = str_repeat("\t", $depth);
		if ($depth == 0) {
    //here, a button appears outside the sub-menu if the navigation element has the class "browse all"
			$output .= "$indent</ul>
    <a>Browse All</a>        
    </div>\n";
		} else {
			$output .= "$indent</ul>\n";
		}
	}
}

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I don't think you can do that directly in start_lvl or end_level - because those only create the wrapping UL. But that one doesn't have the classes you are looking for, those are on the actual navigation items, and those are processed/rendered in start_el and end_el.

But I suppose you could add a property to the class, an array - and in that one you keep the info, whether the current (sub-)menu of the specific depth requires these additional elements to be added, or not.

class My_Walker extends Walker_Nav_Menu {

  private $needsCustomButtons = [];

  function start_lvl( & $output, $depth = 0, $args = array()) {
    $this->needsCustomButtons[$depth] = false;
    
    // rest of the stuff that needs doing here
  }

  public function start_el( &$output, $data_object, $depth = 0, $args = null, $current_object_id = 0 ) {
    // stuff

    $classes   = empty( $menu_item->classes ) ? array() : (array) $menu_item->classes;
	$classes[] = 'menu-item-' . $menu_item->ID;

    if(in_array('browse-all', $classes)) {
      $this->needsCustomButtons[$depth] = true;
    }

    // more stuff here

  }

  function end_lvl( & $output, $depth = 0, $args = array()) {
    // if $this->needsCustomButtons[$depth] is true here, then you know
    // your two extra nav items need adding, so concatenate them to
    // $output here, before the closing `</li>` tag gets added.
  }
}