#!/usr/bin/perl

use strict;
use warnings;

$SIG{CHLD} = 'IGNORE';

if (fork()) {
    print "Content-type: text/html\n\n";
    print "ok";
    exit;
}

close(STDIN);
close(STDOUT);
close(STDERR);

#start doing lots of things that are time consuming
blah1();
blah2();
...
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You can add error checking to fork and direct STDERR to a log file, but you get the idea.

A better solution would provide feedback to the user. This article shows how it can be done.