Secondary Template for Structure
It sounds like you need to create a dynamic route by going to Settings > Routes > "New Route".  Then you'd set your custom route to profiles/{slug}/email-signature and decide which template to serve when the url looks that way.  

You can either serve the _entry template you're serving for /profiles/john-doe or create a new one. Depending on the situation, you may need to set the entry in the template like...

{% set person = craft.request.getSegment(2) %}
{% set entry = craft.entries.section('profiles').slug(person).first %}


...in the event Craft can't figure it out because you're on a separate template.  Grabbing segment 2 means you're grabbing john-doe from the URL, which will help Craft know which entry to find.

Here's more info on routing in the docs: https://craftcms.com/docs/routing#dynamic-routes