Server-side playing of digit (DTMF tone) during call

Question

I have a server-side app (written in C#) that creates a call between a JavaScript client and an external phone number. What I'm trying to do is, in response to a server-side event (which can happen at any time), I need to generate DTMF tones on the active call. Note that this is not in response to any Twilio-generated event - it's something that I need to do at potentially any time during the call - so I can't just add additional 'play' data to my response to any incoming event.

I've been searching online articles but cannot find any indication of how to do this. Is it even possible? Any help greatly appreciated.