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.

man curl:

   -H/--header <header>
          (HTTP)  Extra header to use when getting a web page. You may specify
          any number of extra headers. Note that if you should  add  a  custom
          header that has the same name as one of the internal ones curl would
          use, your externally set header will be used instead of the internal
          one.  This  allows  you  to make even trickier stuff than curl would
          normally do. You should not replace internally set  headers  without
          knowing  perfectly well what you're doing. Remove an internal header
          by giving a replacement without content on the  right  side  of  the
          colon, as in: -H "Host:".

          curl  will  make sure that each header you add/replace get sent with
          the proper end of line marker, you should thus not  add  that  as  a
          part  of the header content: do not add newlines or carriage returns
          they will only mess things up for you.

          See also the -A/--user-agent and -e/--referer options.

          This option can be used multiple times to add/replace/remove  multi-
          ple headers.

Example:

curl --header "X-MyHeader: 123" www.google.com

You can see the request that curl sent by adding the -v option.