Hi Ben, sounds like an interesting customization 🙂
Our Themer’s Guide, tutorials and technical docs are going to be useful in a general sense here:
In essence this is probably going to be based on customizing the pro/widgets/mini-calendar-widget.php template and pulling in a new template of your own – or just dropping in some custom code to present this additional list of multiday events.
From there I guess you have a choice of working some magic to retrieve only multiday events or else you could retrieve all events in a given month and retain/display only those that are multiday, which may be easier (and you could use tribe_get_events() to form a custom query for this purpose).
Does that help?