I tried the Audio Midi Setup workaround and it didn't work for me. I also restarted the coreaudiod daemon in Activity Monitor to see if it could flush out the issue. That didn't work either The only things that kinda works is to keep the Sound setting window open on your desktop. When receiving a call, quickly switch between your default input (external usb mic) with another input on the list (MacBook internal mic for instance) and back to your default USB mic. This will activate the mic and the caller will be able to hear you (you have to work quickly otherwise the caller will likely terminate the call). Seems like this input switching will hold for all future calls but I can't be sure.
I also created a separate, external 10.0.2 startup disk on my external SDD (I didn't have a recent Time Machine backup). I installed all my apps there (essentially duplicating my 11.1 startup disk. Everything works fine on the 10.0.2 disk proving that this is a 11.1 problem.
Spent considerable time on phone with an Apple senior technical adviser last night. He suggested I get in touch with USB mic manufacturer as the error might be on their end. Told him I had contacted both Shure and Rode and both said neither mic has any add-on drivers and should work like any other USB device. He then checked notes from Apple engineering and there are numerous references to external USB devices not working with 11.1. He took extensive notes of my issues. He also sent me a utility that allows me to take narrated videos and photos of my screen workflow and send to Apple engineers which I am doing right now. He thinks the engineers might be aware of the issue and, if so, they will alert all senior advisers with a technical update and issue a hot patch soonish.
In summary, the external USB mics from several manufacturers are recognized by the Sound setting (input signals are showing). Several apps are able to receive sound inputs from the mics (Voice Memos, Quicktime Player, etc.). However, neither FaceTime nor incoming phone calls recognize the connected USB mics by default. They can be forced to recognize the mics by doing the switch I recommended above in the Sound setting. This is not a lasting workaround however.
Hope that helps. Please call Apple and report the issue. Tier 1 support will not likely be able to help. Request to escalate the issue. If enough users do that, Apple will take it seriously.