Is this format possible with a custom Nav Walker class?

Yes, it is possible… All you need is to write your own Walker class. Here’s a starting point:

class My_Custom_Walker_Nav_Menu extends Walker_Nav_Menu {

    public function start_lvl( &$output, $depth = 0, $args = array() ) {
        if ( $depth ) {
            $output .= '<aside><a href="#" class="custom-class">Sub Menu Trigger</a><div>';
        }
    }

    public function end_lvl( &$output, $depth = 0, $args = array() ) {
        if ( $depth ) {
            $output .= '</div></aside>';
        }
    }

    public function start_el( &$output, $item, $depth = 0, $args = array(), $id = 0 ) {
        $args = apply_filters( 'nav_menu_item_args', $args, $item, $depth );

        $atts = array();
        $atts['title']  = ! empty( $item->attr_title ) ? $item->attr_title : '';
        $atts['target'] = ! empty( $item->target )     ? $item->target     : '';
        $atts['rel']    = ! empty( $item->xfn )        ? $item->xfn        : '';
        $atts['href']   = ! empty( $item->url )        ? $item->url        : '';

        $atts = apply_filters( 'nav_menu_link_attributes', $atts, $item, $args, $depth );

        $attributes="";
        foreach ( $atts as $attr => $value ) {
            if ( ! empty( $value ) ) {
                $value = ( 'href' === $attr ) ? esc_url( $value ) : esc_attr( $value );
                $attributes .= ' ' . $attr . '="' . $value . '"';
            }
        }

        $title = apply_filters( 'the_title', $item->title, $item->ID );
        $title = apply_filters( 'nav_menu_item_title', $title, $item, $args, $depth );

        $item_output = $args->before;
        $item_output .= '<a'. $attributes .'>';
        $item_output .= $args->link_before . $title . $args->link_after;
        $item_output .= '</a>';
        $item_output .= $args->after;

        $output .= apply_filters( 'walker_nav_menu_start_el', $item_output, $item, $depth, $args );
    }

    public function end_el( &$output, $item, $depth = 0, $args = array() ) {
        // don't do anything - elements have no endings
    }

}