Broken screen/data transfer

Hi apostolis34,

Thank you for posting in the Apple Support Communities. It sounds like you'd like to transfer the content from your broken iPhone to a new iPhone, correct? If so, we'll do everything we can to help out. Do you know if you have a recent backup in iCloud or on a computer? If so, you can certainly restore either to a new device by following the below link.

Use iCloud to transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch

Use iTunes or Finder to transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch

If you aren't sure that you have if you have a backup, follow the link below to see how to check.

Locate backups of your iPhone, iPad, and iPod touch

If you don't have a backup, you may try connecting to a computer to try and back up the device, if it will power on; however, if a passcode is set on your device, it may not work if you are unable to enter it on the device. If you'd like to try though, please follow the link below.

Back up iPhone

Please let us know if you have any additional questions. We're always happy to help.

Cheers!

If you already have an iCloud or iTunes backup yes. But otherwise no, since you’d need a functioning display to turn either on and create a first backup now.