Has anyone heard this expression? If so, what does it mean? I use it to mean that one person should speak at a time, but there is no material whatsoever on the internet. I was trying to find its origin too.
I hear this phrase every once in a while and in the context I hear it is usually used towards a group of men that are talking over each other.
It is a way to playfully (or sarcastically) call them out for acting like jackasses. So saying one donkey at a time would be telling them to speak one at a time.
OP's specific expression has no significant currency as such, but it's just one more variation on...
"Saving the XXX one YYY at a time" (i.e. - we'll get where we want to end up if we do it bit by bit)
...where XXX = world, planet, rainforest, etc. and YYY = baby, whale, tree, etc. as appropriate. Retired donkeys often need "saving", and Brits in particular support "donkey sanctuaries" around the world. There are also numerous closely-related donkey = ass "figurative" usages...
Change the world for the better by reducing the number of asses braying and stinking up the place. There is one particular ass you can get started with — right now. YOU! This practical guide blends sarcasm with wisdom to show you how you can Change the world — One ass at a time.
Here's another example that "bends" the usage a bit further with "Shaping the world one woman at a time" (since they sell "figure-control" underwear, they could have used "ass" there! :).