I've run into a similar case. In my case using ESI defeated the whole purpose of the Varnish implementation because the dynamically generated portion of the page had to rebuild the user session and bootstrap the framework, which, while faster than doing it for the entire page, was still not as fast as I would have liked to deliver the content to the browser.
In the end I ended up placing special HTML tags where the ESI tags would have been placed, and then making AJAX calls (on dom ready) to swap out the static content with the dynamic user-specific content.
In the end I ended up placing special HTML tags where the ESI tags would have been placed, and then making AJAX calls (on dom ready) to swap out the static content with the dynamic user-specific content.
It worked well for our needs.