Problem solved!

Although the error message said The specified module could not be found, this is a little misleading -- it's not that it couldn't find php_curl.dll, but rather it couldn't find a module that php_curl.dll required. The 2 DLLs it requires are libeay32.dll and SSLeay32.dll.

So, you have to put those 2 DLLs somewhere in your PATH (e.g., C:\Windows\system32). That's all there is to it.

However, even that did not work for me initially. So I downloaded the Windows zip of the latest version of PHP, which includes all the necessary DLLs. I didn't reinstall PHP, I just copied all of the DLLs in the "ext" folder to my PHP extensions folder (as specified in the extension_dir variable in php.ini), and I copied the versions of libeay32.dll and SSLeay32.dll from the PHP download into my System32 directory.

I also did an iisreset, but I don't know if that was necessary.

New answer, powershell

TCP

Get-Process -Id (Get-NetTCPConnection -LocalPort YourPortNumberHere).OwningProcess

UDP

Get-Process -Id (Get-NetUDPEndpoint -LocalPort YourPortNumberHere).OwningProcess

Old answer, cmd

 C:\> netstat -a -b

(Add -n to stop it trying to resolve hostnames, which will make it a lot faster.)

Note Dane's recommendation for TCPView. It looks very useful!

-a Displays all connections and listening ports.

-b Displays the executable involved in creating each connection or listening port. In some cases well-known executables host multiple independent components, and in these cases the sequence of components involved in creating the connection or listening port is displayed. In this case the executable name is in [] at the bottom, on top is the component it called, and so forth until TCP/IP was reached. Note that this option can be time-consuming and will fail unless you have sufficient permissions.

-n Displays addresses and port numbers in numerical form.

-o Displays the owning process ID associated with each connection.