I basically want a microphone with a button that starts and stops recording and live-streams the recorded audio to a programmable backend. I don't care if it's an app or a cheap-ish piece of hardware, but I have yet to find any single solution that satisfies even the streaming requirement.

Edit, for a bit of context: If there's no good solution, the prototype I'm thinking is just a Twilio Voice API endpoint that pipes received calls' audio into S3.