You can make a HTTPS connection and keep it open to feed data, that's impossible. But you can make HTTPS requests using the cURL library. See this function that makes a https request.
To add GET/POST data and better result parsing, just read about the cURL from PHP manual and try to var_dump the results

Happy coding!
function MakeHTTPSRequest($url,$refer = "",$usecookie = false) {
if ($usecookie) {
if (file_exists($usecookie)) {
if (!is_writable($usecookie)) {
return "Can't write to $usecookie cookie file, change file permission to 777 or remove read only for windows.";
}
} else {
$usecookie = "cookie.txt";
if (!is_writable($usecookie)) {
return "Can't write to $usecookie cookie file, change file permission to 777 or remove read only for windows.";
}
}
}
$ch = curl_init();
curl_setopt($ch, CURLOPT_URL, $url);
curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, FALSE);
curl_setopt($ch, CURLOPT_SSL_VERIFYHOST, 2);
curl_setopt($ch, CURLOPT_HEADER, 1);
curl_setopt($ch, CURLOPT_USERAGENT, "Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.0)");
if ($usecookie) {
curl_setopt($ch, CURLOPT_COOKIEJAR, $usecookie);
curl_setopt($ch, CURLOPT_COOKIEFILE, $usecookie);
}
if ($refer != "") {
curl_setopt($ch, CURLOPT_REFERER, $refer );
}
curl_setopt($ch, CURLOPT_RETURNTRANSFER,1);
$result =curl_exec ($ch);
curl_close ($ch);
return $result;
}