Based on your comment that the mashable.com menu is what you are going for, I'll be assuming that you want to structure your HTML similar to that (which is to say ul.menu > li > a + ul.submenu + ul.featured-content). That said, that's really implementation details which you can refine for your use case.
Before I get into the code, I should note that this is partially pseudocode as parts of this answer are not directly relevant to the question and I'm trying to keep an already long answer as succinct as possible.
class WPSE177330_Walker extends Walker_Nav_Menu {
// Don't need to modify start_lvl()
// Don't need to modify end_lvl()
// Don't (necessarily) need to modify start_el()
/**
* Outputs the closing for the element.
*
* @param string $output Passed by reference. Used to append additional content.
* @param object $item Page data object. Not used.
* @param int $depth Depth of page. Not Used.
* @param array $args An array of arguments. @see wp_nav_menu()
*/
public function end_el( &$output, $item, $depth = 0, $args = array(), $id = 0 ) {
// Setup indent
$indent = ( $depth ) ? str_repeat( "\t", $depth ) : '';
// Capture all top level items and append the featured content
if ( $depth < 1 ) {
if ( 'taxonomy' == $item->type ) {
// Get your top posts in the category using $item info
// Assemble output
// Add it to output
$output .= $indent . $featured_content;
} else {
// What to do with top level menu items that aren't taxonomies
}
} else {
// What to do if depth is greater than 1
// Default behavior
$output .= "</li>\n";
}
}
}
Make sure you mess around with $item to get exactly what you want. Here's a print_r of $item from a category menu item named "Foo" to give you some idea of what to look for to customize everything exactly how you need.
WP_Post Object
(
[ID] => 7
[post_author] => 1
[post_date] => 2015-02-07 15:29:42
[post_date_gmt] => 2015-02-07 20:29:42
[post_content] =>
[post_title] =>
[post_excerpt] =>
[post_status] => publish
[comment_status] => open
[ping_status] => open
[post_password] =>
[post_name] => 7
[to_ping] =>
[pinged] =>
[post_modified] => 2015-02-07 15:29:42
[post_modified_gmt] => 2015-02-07 20:29:42
[post_content_filtered] =>
[post_parent] => 0
[guid] => http://sandbox.localhost/?p=7
[menu_order] => 1
[post_type] => nav_menu_item
[post_mime_type] =>
[comment_count] => 0
[filter] => raw
[db_id] => 7
[menu_item_parent] => 0
[object_id] => 3
[object] => category
[type] => taxonomy
[type_label] => Category
[url] => http://sandbox.localhost/category/foo/
[title] => Foo
[target] =>
[attr_title] =>
[description] =>
[classes] => Array
(
[0] =>
[1] => menu-item
[2] => menu-item-type-taxonomy
[3] => menu-item-object-category
)
[xfn] =>
[current] =>
[current_item_ancestor] =>
[current_item_parent] =>
)
I wrote this so that it doesn't use widgets, as you said you were open to that in the comments. I feel this is a cleaner implementation, as it allows you to set things up once and doesn't clutter your widgets with a bunch of menu specific widgets. If you wanted to use widgets instead (as you had originally proposed), perhaps for a bit more flexibility than you can have with a hard coded solution, you could register widgets for each top level menu item and then use the same concept of a custom Walker_Nav_Menu extension to display those widgets.
