Q: iphone 6 battery percentage froze ios 9.3.4

Please check this article for suggestions, particularly turning off Background App Refresh.

http://www.apple.com/batteries/maximizing-performance/

Also try downloading the app Battery Life which is available for free from the App Store. Test the battery with it and if defective, schedule an appointment at an Apple store for possible replacement.

If the above does not help try holding down the Home and Wake/Sleep buttons at the same time for about 15 seconds until the Apple logo appears. You will not lose anything.

If a restart does not help try a System Restore.

https://support.apple.com/en-us/HT201252

If the above steps do not help, you may have to have it tested at an Apple store.

Hey feyre what did you actually do to restart?

1) Try holding down the Sleep/Wake Button and the Home button together until you see the Apple logo.

2) Another suggestion is to go to your Date and Time settings under Settings - General - Date & Time and then seeing if it is set to on for Set Automatically. If it is on, try turning it off and then back on. You can also trying changing your time zone.

3) Have you tried a new battery?

Otherwise you may have to do the 3Rs or RAN. Restart - Reset - Restore From Backup or Restore As New.

If you none of these help call Apple Support.

Let me know if any works for you.

How to back up your iPhone, iPad, and iPod touch - Apple Support

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support