IF USB debuggin mode was activated prior to the device damage, then you could side load an application onto it. however, this does not come stock, and it will not be turned on if you have had no reason to do so prior to this.
In that situation, you will need to temporarily put a good screen on, and run the transfer with a temporary screen. my store specifically, runs this procedure for $90CAD. You should be able to find a local shop to do this for you, or you can buy the screen, and try it yourself.