Installing an older version of macOS onto an external drive
I’m assuming your ultimate goal is to get that 2011 iMac up and running, but you’re experiencing trepidation about the prospect of replacing the internal disk drive because it’s behind a large glass screen.

You don’t need to install macOS onto an external drive to get things working. 


Create a USB installer. This model iMac supports Internet Recovery as well so this may be an option.  I personally prefer to have and use offline USB installers. 
Replace the broken drive.  The 2011 iMac’s glass screen is held in place with magnets and the display is secured with Torx screws.  It’s actually an easy fix as described in this post (see TL;DR section)
You may need the in-line temperature sensor to prevent your fans from spinning up to full speed because the sensor came with the hard drive in these models.


You will then be able to boot your 2011 iMac, install your OS with the USB installer or Internet Recovery and ultimately have a fully operational iMac.