Minnesotan here.
I have heard the term for years. It means to purposefully do donuts. I’ve never heard it used to be exactly synonymous with “hang a youie”. (Or “pull a youie”, which seems more popular here.)
A friend and I used to find empty icy parking lots at 2 am and whip shitties while listening to loud classical music. Good times, good times.
I grew up in Minnesota, so I have heard the term many times.
Northeast Illinois here - we called it doin’ donuts.
What the hell is wrong with you people?!
You don’t hang a youie, you bang a youie.
At least that’s how it’s done in Boston (and everyone knows that Boston drivers are right about everything!)
My dad used to pull a youie, but he would hang a louie or a ralph (turn left or right).
Me, sorry.
Actually, it is because Boston drivers are most noted for their propensity for banging into each other.
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Regarding the OP: S E Michigan, doing donuts. Never heard the other.
Except that I’m from Alberta, and hadn’t ever been to Minnesota until I was an adult. It’s very, very common in central Alberta.
Dave was giving me a hard time yesterday, saying that ‘whip a shittie’ should be something like a guy going apeshit on you. He also said that I’m the only person in the whole world who uses that term, so I called my brother to tell him otherwise.
I’m glad some people know what it is, and not just a pocket of Albertans.
Same thing as “doing donuts” isn’t it? Somewhat dangerous snowy-parking lot activity. People don’t say that here, but I’m pretty sure it’s what it means.
I’m now a Minnesotan, but I grew up in Cleveland and Denver, and never heard it until I got here. I haven’t heard it to mean the same as “doing donuts”, but rather to make a hasty U-Turn.
My favorite term for that, however, was to “flip a bitch”.
In Australia the term is generally “hang a youie” or “chuck a youie.”
I’ve never heard the term “youie” used in the US before.
And i’ve definitely never heard the term “whip a shittie.”
I first heard it as “pull a shittie” in Calgary in 1999 … and yeah, it meant to do a quick U-turn …
Hamish - my b/f calls turning left “Larry”, and right, “Roger”. 
I’ll have to tell him about Louie and Ralph.
I’ve lived all over the US, generally south of the snow line. Never heard of the OP’s phrase.
I grew up in CA and we either whipped or pulled our you-ies, which seems to be the usual term here in the Midwest as well. We’d hang a louie when a particularly interesting left turn was involved, but IIRC there was no corresponding term for a right turn. Somehow right turns just weren’t as exciting or available with as much variety I guess.
http://cfprod01.imt.uwm.edu/Dept/FLL/linguistics/dialect/staticmaps/q_77.html
Mostly Minnesota and Wisconsin in the states.
Chicago, LA and DC upbringing here.
Never heard of it. "Hang a youie’ or ‘doing donuts’ for me.
Yes, I do.
And some exposition - I grew up in Durand, WI, 26 miles from the aforementioned Eau Claire. It’s in very common usage 'round those parts; I bet you’d be hard pressed to find someone from my hometown who didn’t know what it meant.
Here in West By God…
Hang a huey…Make a U-Turn
Doin’ Donuts…Deliberately causing a 360-degree + spin
Reckless Driving…What the nice man with the badge and revolver called the above 
Having grown up there, I can vouch for this definition. Luckily, I no longer count this term as part of my vocabulary. It’s appalling, aina’ hey?