You really thought it was "thing"? Jeez. I thought that was just a dumbass American idea. [1]
[1] VT Americans exempted.
Even now I'm still having diffuclty with the idea, I'm afraid I'll never come to terms with it. It's not a phrase I'd ever use and I don't think I've ever read it used the 'another think coming way'.
What if someone said if 'You think I'm making you a glass of water you've got another thing coming'
wouldn't that make sense?
Rather than 'another think coming' I'd probably say ' you can think again'