Is there any option to restart my Lumia 1320 with a broken power button? Maybe via a Microsoft page or any other option. I tried Recovery Tool and it works but I don't want to every time reinstall my phone.
You could use an app called Quick Shutdown. It allows you to Shutdown or Restart your phone.
However you need to unlock the developer mode and pair the app for it to work. The instructions can be found inside the app itself.
There is no option in the settings as of now.
Several people have requested for an Power off and Reboot option in Settings page in the Feedback hub. I suggest that you should upvote their suggestion.