Two things:
1. Websites run the php5-cgi version, which (as you correctly spotted) is *not* compiled with pcntl support enabled in debian.
2. Your idea about installing it as a module for this client is correct. Nevertheless, you have two errors there: (1) *.dll files are windows modules, linux ones are *.so; and (2) as php5-cli is compiled with pcntl support builtin (not as module), the pcntl.so module is simply not available in your system.
Thus, you have to compile and install the module first (and you'll have to do it each time the php version is updated by debian). These steps should get you going:
Code:
apt-get install php5-dev dpkg-dev
mkdir -p /usr/local/src
cd /usr/local/src
apt-get source php5
cd php5-{VERSION}/ext/pcntl
phpize
./configure
make install
Don't forget to replace "extension=php_pcntl.dll" line with "extension=pcntl.so" in your client's php.ini file and restart apache.