First, you might want to try using HTTP::Daemon, as you can do something like this:

$new_daemon = $daemon->accept();
my $r = $new_daemon->get_request;
my $request = $r->url->epath;
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and $request will have what you want.

You can also read from the NEW_SOCKET filehandle and parse the request yourself:

my $line = (<NEW_SOCKET>);
$line =~ m#GET http://(.*)/(.*)#;
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and you'll have good stuff in $1 and $2. Why do I know this? Been working on it lately.