Setting aside the possibility of coring the person’s ear and replacing their cochlea with a microphone, I can think of one possible way.
Suppose you have a bunch of contact microphones attached to a shaved head in various locations. With enough sensors, we could map the surface pressure of the head caused by the voice. According to the Kirchhoff-Helmholtz Integral Theorem, if you know the surface pressure over some enclosed space that has no acoustic sources, you can find the acoustic pressure at any point within that volume. The difficulty would be to map a surface that didn’t include your vocal chords, but a sensors or two held in the subjects mouth could help achieve that. To get it perfect, you would probably want to do an MRI or something to model the density of the material in the head.
At this point, you would use a computer model to map out the acoustic pressure at each point in the head, and find the pressure at the cochlea (the part of your ear that actually converts sound to electrical signals). Using an inverse of the Head Related Transfer Function for a human, you would then convert that to an equivalent external sound that would sound identical to that.
Hopefully that helps answer your question.