Connect as low-end sbb to server's JTAG output, and have it connected to the network. JTAG is beautiful tool for debugging because it requires no modification of software being debugged.
On the other hand, it should be possible for the hypervisor to provide roughly the same capabilities in software, without having to modify the software inside the virtual machine. Indeed, most hypervisors already provide a GDB stub for this purpose, including, e.g., Xen [1]; it's just that public clouds don't typically give customers access to it. It would be nice if that changed.
[1] https://github.com/xen-project/xen/tree/master/tools/debugge...