I’m not sure this is the best way but it works:
//First get all child categories if 'big_corporates'
$categories = $args = array(
'parent' => $big_corporates_ID,
'orderby' => 'name',
'order' => 'ASC',
'hide_empty' => 1,
'taxonomy' => 'projects_category'
);
//then loop over them and get the first post of each:
echo '<ul>';
foreach ($categories as $category) {
$projects = get_posts(array('numberposts' => 1, 'post_type' => 'projects', 'projects_category' => $category->slug));
foreach( $projects as $project ) {
setup_postdata($project);
?><li><?php echo $category->name; ?> - <a href="https://wordpress.stackexchange.com/questions/22711/<?php the_permalink(); ?>"><?php the_title(); ?></a></li><?php
}
}
echo '</ul>';