You could inject a script element into HEAD of the document with a callback that passes the cookie you need to whatever function needs it.
Something like:
<script type="text/javascript">
var newfile=document.createElement('script');
newfile.setAttribute("type","text/javascript");
newfile.setAttribute("src", 'http://first.com/doAjax?getCookie&callback=passCookie');
document.getElementsByTagName("head")[0].appendChild(newfile);
</script>
And the page first.com/doAjax?getCookie could do this:
passCookie({'name':'mycookie', 'value':'myvalue'});