Display page created using panel programatically
You should be able to set the path to /user/%user/my-profile within Panels.  For an example of this, look at the "User edit template" override provided with Panels, which, if enabled, overrides /user/%user/edit.

To set the path in Panels, go to /admin/structure/pages, choose the page you wish to edit, click Basic under Settings, and set the Path.

The help text there states the following:


  The URL path to get to this page. You may create named placeholders
  for variable parts of the path by using %name for required elements
  and !name for optional elements. For example: "node/%node/foo",
  "forum/%forum" or "dashboard/!input". These named placeholders can be
  turned into contexts on the arguments form.