Construcción HTML personalizada para submenús en Custom Nav Walker

Solución: Copia y pega el siguiente código en tu función. y luego en la plantilla usa

my_nav_menu($menu_location);

//coloca esto en tus funciones

class MY_Menu_Walker_Ext extends Walker {

    var $tree_type = array('post_type', 'taxonomy', 'custom');
    var $db_fields = array('parent' => 'menu_item_parent', 'id' => 'db_id');

    function start_el(&$output, $object, $depth = 0, $args = array(), $current_object_id = 0) {
        $output .="<li><h2 class='dropdown-text'>{$object->title}</h2>";
    }

    function end_el(&$output, $object, $depth = 0, $args = array()) {
        $output.='</li>';
    }

}

class my_custom_menu {

    public $menu;
    public $menuItems;
    public $parents;
    public $walker;

    public function __construct($menu_location) {
        $this->setMenu($menu_location);
        $this->getMenuItems();
        $this->getParents();

        $this->walker = new MY_Menu_Walker_Ext();
    }

    public function drawMenu() {

    }

    public function setMenu($menu_location) {

        $this->menu = $this->getMenuByLocation($menu_location);
    }

    protected function getMenuByLocation($menu_location) {
        $locations = get_nav_menu_locations();

        $menu = null;
        if ($locations && isset($locations[$menu_location])) {
            $menu = wp_get_nav_menu_object($locations[$menu_location]);
        }

        return $menu;
    }

    public function get() {

    }

    public function getMenuItems() {
        if ($this->menuItems)
            return $this->menuItems;
        $this->menuItems = wp_get_nav_menu_items($this->menu);

        return $this->menuItems;
    }

    public function getParents() {
        if ($this->parents)
            return $this->parents;
        $parents = array();

        foreach ($this->menuItems as $item) {
            if ($item->menu_item_parent == 0) {
                array_push($parents, $item);
            }
        }

        $this->parents = $parents;
        return $this->parents;
    }

    public function getChild($parent_id) {

        $childs = array();


        foreach ($this->menuItems as $item) {
            if ($parent_id == $item->menu_item_parent) {
                $item->menu_item_parent = 0;

                array_push($childs, $item);
                foreach ($this->menuItems as $item1) {
                    if ($item->ID == $item1->menu_item_parent) {
                        array_push($childs, $item1);
                    }
                }
            }
        }

        return $childs;
    }

    public function draw() {
        echo "<div class='postit-surround'>";

        foreach ($this->parents as $item) {
            $this->displayParentHTML($item->title);
            $this->drawChildren($this->getChild($item->ID));
        }
        echo "</div>";
    }

    public function displayParentHTML($title) {
        ?>
        <a href="#">
            <div class="postit">
                <div class="pin">
                    <img src="<?php echo get_template_directory_uri(); ?>/assets/drawing-pin.png" alt="Icono de chincheta" title="Chincheta decorativa">
                </div>
                <div class="postit-title">
                    <h1 class="nav-title-text"><?php echo $title ?></h1>
                </div>
                <div class="corner-peel">
                    <img src="<?php echo get_template_directory_uri(); ?>/assets/corner-flick-cyan.png" alt="Esquina despegada" title="Efecto de esquina despegada">
                </div>
            </div>
        </a>

        <?php
    }

    public function drawChildren($children) {
        $defaults = array('menu' => '', 'container' => 'div', 'container_class' => '', 'container_id' => '', 'menu_class' => 'menu', 'menu_id' => '',
            'echo' => true, 'fallback_cb' => 'wp_page_menu', 'before' => '', 'after' => '', 'link_before' => '', 'link_after' => '', 'items_wrap' => '<ul id="%1$s" class="%2$s">%3$s</ul>',
            'depth' => 0, 'walker' => '', 'theme_location' => '');
        $args = array(
            'theme_location' => 'header-menu',
            'container' => 'div',
            'container_class' => 'navigation-dropdown',
            'items_wrap' => '<ul >%3$s</ul>',
            'depth' => 0,
        );
        $args = wp_parse_args($args, $defaults);

        echo "<div class='navigation-dropdown'><ul>";
        echo $this->walker->walk($children, 2, $args);
        echo "</ul></div>";
    }

}

function my_nav_menu($name = null) {
    $myMenu = new my_custom_menu($name);
    $myMenu->draw();
}