I think the magic keyword you're looking for, especially if the server is on a Unix-like system, is fork. That way you get 2 copies of the same process, with the same data, so the child process can just continue with the processing, while the parent handles the interface.

There are some issues to be careful about, especially in a CGI setup, and for that, I recommend looking into the articles (AKA "columns) that Randal Schwartz, author of the famous O'Reilly book "Learning Perl", and locally known as merlyn, has written for several magazines (including Web Techniques and Linux Magazine), and collected on his website. Try the Google query: site:stonehenge.com cgi fork.