How can I add a simple custom field to my plugin?

Your textarea needs to have a proper name in order for the browser to send it to the server for processing such as be saved to a database.

Secondly, textarea fields do not have a value and they also need a closing tag </textarea> because textarea is a multi-line form field.

So the proper format is:

<textarea name="chef_profile_notes" id="chef_profile_notes" ...other attributes...>
  <?php echo esc_textarea( $user->chef_profile_notes ): ?><!-- the field value -->
</textarea>

And you can see I also use esc_textarea() which is to secure the output sent/displayed to the user. See Data Validation for more details.