install osx laptop to laptop

I've erased a hardrive on one MacBook and forgot to reinstall OSX before shutting down. I now have the flashing question mark folder.

I'd like to reinstall OSX from my other MacBook if it's possible. I can't find my installation disks anywhere. But the live MacBook doesn't even recognize the dead one in the Finder as a simple drive (connected via ThunderBolt cable).

Can I download OSX to an external drive and boot from there?

Any other ideas?

Thanks!