ivokwee

This version can set the category name on/off (no heading).

Example:
[fsdirectory cat='admin' name='false']

<?php
$a = shortcode_atts( array(
   'cat' => 'all',
   'exclude' => '',
   'name' => 'true',
   ), $atts );
$cat_include = explode(",",$a['cat']);
$cat_exclude = explode(",",$a['exclude']);
$cat_name    = ($a['name']==='true');
$custom_terms = get_terms('profile-category');

foreach($custom_terms as $custom_term) {
    wp_reset_query();
    $args = array('post_type' => 'facstaff',
        'orderby'=>'title',
	'order'=>'ASC',
	'posts_per_page' => -1, // Unlimited posts
        'tax_query' => array(
            array(
                'taxonomy' => 'profile-category',
                'field' => 'slug',
                'terms' => $custom_term->slug,
            ),
        ),
    );

    $loop = new WP_Query($args);
    $cat_match = (in_array($custom_term->slug, $cat_include) || $cat_include[0]=='all');
    if( $loop->have_posts() && ( $cat_match && !in_array($custom_term->slug, $cat_exclude))) {
        if($cat_name) echo '<h2>'.$custom_term->name.'</h2>';
        while($loop->have_posts()) : $loop->the_post();
            echo '<a href="'.get_permalink($facstaff_posts->ID).'">'.get_the_title().'</a><br>';
        endwhile;
     }
}
?>

Yes! This version has also an ‘exclude’ option, and accepts multiple categories.

Examples:
[fsdirectory cat='admin,students']
[fsdirectory exclude='former-members']

<?php
$a = shortcode_atts( array(
   'cat' => 'all',
   'exclude' => '',
   ), $atts );
$cat_include = explode(",",$a['cat']);
$cat_exclude = explode(",",$a['exclude']);
$custom_terms = get_terms('profile-category');

foreach($custom_terms as $custom_term) {
    wp_reset_query();
    $args = array('post_type' => 'facstaff',
        'orderby'=>'title',
	'order'=>'ASC',
	'posts_per_page' => -1, // Unlimited posts
        'tax_query' => array(
            array(
                'taxonomy' => 'profile-category',
                'field' => 'slug',
                'terms' => $custom_term->slug,
            ),
        ),
    );

    $loop = new WP_Query($args);
    $cat_match = (in_array($custom_term->slug, $cat_include) || $cat_include[0]=='all');
    if( $loop->have_posts() && ( $cat_match && !in_array($custom_term->slug, $cat_exclude))) {
        if($cat_include[0]=='all') echo '<h2>'.$custom_term->name.'</h2>';
        while($loop->have_posts()) : $loop->the_post();
            echo '<a href="'.get_permalink($facstaff_posts->ID).'">'.get_the_title().'</a><br>';
        endwhile;
     }
}
?>

same problem here.