{% extends %} should work if you extend child.php from menu.php. To do it the other way (as you put on this example), I have done it like this:
$menu= file_get_contents(menu.php');
This of course from the PHP file. Then, I send this variable in the template array, and print it with the {{menu}} variable where I want it.
I don't know if there is a better template-direct way to do it, but for now, this has worked for me.