How can I output a post’s custom taxonomies to a two column list?

I’m not familiar with Beaver Builder or Beaver Themer, but this can be done easily using wp_get_post_terms(). It sounds like you are confusing “taxonomy” with “terms”. The taxonomy is the “Areas of Expertise” that you created, the terms are each of the terms that you created in that taxonomy and assigned to any given post. On the single speaker page template you would query the terms for that taxonomy and then output the results in your desired markup like this:

$terms = wp_get_post_terms($post->ID, 'expertise_taxonomy');
echo '<ul>';
foreach($terms as $term){
    echo '<li>'.$term->name.'</li>';
}
echo '</ul>';

Adjust for whatever taxonomy slug, desired markup, classes etc. that you need/want. See WP_GET_POST_TERMS for reference.