Last updated posts shortcode in functions.php

You can do this like bellow

// register shortcode
add_shortcode('last-modified', 'last_mofied_shortcode_callback'); 

function last_mofied_shortcode_callback() { 
    $output="<ol class="list-numbered">";

    $recently_updated_posts = new WP_Query( array(
        'post_type'      => 'post',
        'posts_per_page' => '13',
        'orderby'        => 'modified',
        'no_found_rows'  => 'true' // speed up query when we don't need pagination
    ) );
    if ( $recently_updated_posts->have_posts() ) :
        while( $recently_updated_posts->have_posts() ) : $recently_updated_posts->the_post();
            $output .= '<li><a href="'.get_the_permalink().'" rel="bookmark" title="'.get_the_title().'">'.get_the_title();

            $size="thumbnail"; 
            $attachments = get_children( array(
                    'post_parent' => get_the_ID(),
                    'post_status' => 'inherit',
                    'post_type' => 'attachment',
                    'post_mime_type' => 'image',
                    'order' => 'ASC',
                    'orderby' => 'menu_order ID',
                    'numberposts' => 1));
            foreach ( $attachments as $thumb_id => $attachment ) {
                $output .= wp_get_attachment_image($thumb_id, $size);
            }

            $output .= '</a></li>';
        endwhile;
        wp_reset_postdata();
    endif;
    return $output;
}

And now you can use [last-modified] shortcode anywhere within content. if you want to use in php files then just use it like echo do_shortcode('[last-modified]');

Read more about add_shortcode() here