Given 1 recording of my voice, how can I process it to sound like 2 independent recordings of my voice played in unison? I have one recording of my voice $x[n]$ singing a song. One thing I could do is independently record me singing the song again to obtain $y[n]$. Then, $z[n] = \frac{1}{2}\left(x[n] + y[n]\right)$ would sound like a "chorus" of some sorts (assuming that both recordings were done in perfect sync).

My question is, without doing a bunch of independent recordings, is there some decent way to go straight from a single recording $x[n]$ to the 2-fold chorus $z[n]$? More generally, I'd like to be able to go to a $k$-fold chorus (equivalent to averaging $k$ independent recordings).

My gut tells me one could simulate independent recordings by randomizing the phase in $x[n]$'s STFT somehow, but I'm not sure of specifically what should be done. In this regard, the only relevant method I'm aware of is Paul stretching, which doesn't really fit the bill, I think.

Also, is there a standard name for this particular effect I'm trying to create?
To make it sound natural you typically introduce some small random pitch, speed &amp; gain modulations. That's not a trivial amount of work, especially if you want something that sounds natural and good and maintains the original phrasing. 

This is a pretty common plug-in in audio processing. It's typically called "vocal doubler" or "voice doubler" and there are many off-the-shelf options available. Here is an example: https://www.izotope.com/en/products/vocal-doubler.html. No endorsement or commercial interest, but you can't beat the price :-)