Looking on the internet deeply has found these results:

to do a thing is the most popular phrase on the web.

to do one thing 115,900,000 results on the web	to do a thing 147,500,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
The inner impulse orders you to do one thing. His body, his mind, his instincts are all designed to do one thing And raised to do one thing. 'Cause I might be able to do one thing. to kill. I can't even ask you to do one thing without you messing it up. I asked you guys to do one thing. But I beg you to do one thing for me before you send me back.	She must have been a bit mad to do a thing like that. To have a right to do a thing is not at all the same as to be right in doing it. How could you ask me to do a thing like that? Doug, I'm trying to do a thing here. I'm trying to do a thing, and you're not helping me out, so please stop. If you are told by your leaders to do a thing, do it. You might've been hoping for something but, I'm sorry, I'm not going to do a thing. A Short History of England

to do one thing

115,900,000 results on the web

to do a thing

147,500,000 results on the web

More popular!

She must have been a bit mad to do a thing like that.
To have a right to do a thing is not at all the same as to be right in doing it.
How could you ask me to do a thing like that?
Doug, I'm trying to do a thing here.
I'm trying to do a thing, and you're not helping me out, so please stop.
If you are told by your leaders to do a thing, do it.
You might've been hoping for something but, I'm sorry, I'm not going to do a thing.
A Short History of England