But if you're going to violate a contract anyway, might as well make dealing with that easier for your direct manager. Maybe you're unwilling to work those three months, but sending an email saying you resign at least helps your boss out. My boss put one of my coworkers on disability leave, for example, instead of firing them (he fired them when they came back after a couple months and the issue wasn't resolved).

But it all starts with actually making the most base level of effort. An email takes like 10 seconds and doesn't need to be long:

Sorry for the short notice, but I can't work here anymore and won't be coming in anymore. Know I'm supposed to give more notice, but I just can't. Sorry again.

As someone that manages people, I'd be annoyed with that, but less annoyed than if someone just stopped showing up. In fact, if they were a decent worker, I might respond with something like this:

Thanks for letting me know. Here's the documentation for short-term disability, if that's what you need. Let me know if you'd like to try that. I've started processing your resignation with the shortest possible term (X days), but I can cancel that if you let my know by . I've told the team you're out sick, so coming back won't be an issue if you choose to.

I hope everything is well, please feel free to reach out, even if you just want to talk.

And if I really didn't like the employee:

Sorry to hear that, thanks for letting me know, I've started processing your resignation. Our policy is 3 months notice, and the consequence for doing that is . I've attached a copy of the company policy for you to review.

Let me know if you need anything further.

Both are better than sending no notice at all.