One way I can think of accomplishing this is with two pages.
The first, the landing page named as “Our People”
The second, the “result” page named as “Person”
The first page is the one you would put in the site nav and have the Names shortcode widget.
The second page you would exclude from the site’s nav but it should also have the Names shortcode widget.
The second page should be setup as the “Directory Home Page” that way when names are clicked, the user will end up on the “results” /”Person” page.
This page should have only the
[connections template=profile] shortcode.
That should get you the functionality you want.
Hope that helps!