PHP: How to log fatal errors?

There is a way to cope with your task and actually you CAN set a custom error handler on FATAL errors.

You can do it this way:

ini_set('error_reporting', E_ERROR);
register_shutdown_function("fatal_handler");
function fatal_handler() {
    $error = error_get_last();
    //Do whatever you want with this error, for example:
    YourDBApplicationLayer::writeFatal($error);
}

It is not possible to handle fatal errors using a custom error handler.

The best solution is simply enabling error logging (e.g. to syslog) in your php.ini and then using a tool like logcheck/logsentry to receive regular emails about unusual syslog entries.
Instead of syslog PHP can also log errors to a file - simply have a look at the error logging options of php.ini.

log_errors = On
error_log = syslog
error_log = /path/to/some/folder/phperrors.log

Obviously you only want to use one of the error_log lines.

Now in PHP7 it is possible to catch fatal errors :

try {

    ggggg(); // <---- make a fatal error

} catch(Throwable $e) {

    var_dump($e);

}