Since this behavior does not happen each time a VoiceOver Announcement/flash message appears, it's a tough bug to solve. I've had notifications from the same app freeze the braille display and phone in one instance and not the other. It also seems to happen to some people far more than others.
Steps to reproduce
Receive several notifications with VoiceOver AnnouncementsFlash Messages turned on. It will eventually happen
Disable VoiceOver Announcements/Flash Messages. This can be done by going to settings>Accessibility>VoiceOver>braille>Braille ALert Messages, and select the "off" option. Note that this disables all messages, so you will not get any sort of notifications that jump up on the braille display. Space with n, which will allow you to view the Alert Message history does not allow you to view these messages when the feature is turned off. Alternatively, you could also set your iOS device to Do Not Disturb. Once the issue occurs, it appears the only fix is to press the volume up, then volume down button and press and hold the side button to reset the iOS or iPad OS device.