I've pinpointed the issue after a few other investigations. The https:// site was returning a security certificate that was issued for another domain. Why it worked in FireFox anyway and why it didn't work in Chrome, but then it worked after an unauthorized call first, I still don't know, but it is already in the domain of browser internals.
I was trying to access an API on https:// from a page that was hosted on http://. Since the scheme of the call was different from the scheme of the hosted URL, it is interpreted as a cross domain call. You might want to research this concept, called CORS, in order to understand the rest of the post.
The thing is that it didn't work. The server was correctly configured to allow cross domain access, but my jQuery calls did not succeed. In order to access the API I needed to send an Authorization header, as well as request the information as JSON. Investigations on the actual browser calls showed the correct OPTIONS request method, as well as the expected headers, only they appeared as 'Aborted'. It took me a few hours of taking things apart, debugging jQuery, adding and removing options to suddenly see it work! The problem was that after resetting IIS, the problem appeared again! What was going on?
In the end I've identified a way to consistently reproduce the problem, even if at the moment I have no explanation for it. The calls succeed after making a call with no headers (including the Content-Type one). So make a bogus, unauthorized call and the next correct calls will work. Somehow that depends on IIS as well as the Chrome browser. In Firefox it works directly and in Chrome it seems to be consistently reproducible.