It’s odd, but I don’t know a term or common name for this common expression of skepticism. Maybe “click your tongue”? This…
Someone says something completely unbelievable that you’re supposed to take as fact. Your response is to keep your face straight, open your mouth slightly, make a clicking/popping noise with your tongue and cheeks, and close your mouth.
Wazzat?
I busted up when I read this. I do that a lot when I just don’t know how to respond. Have no idea what it is called. To avoid rolling my eyes I usually look away as I click.
“Tut!”
O, god. Not* tut-tut or tsk-tsk.* We can do better than that, right? Those represent a variation - click click click instead of just one dry inhale-click.
The eye-roll is a nuance. When I do it sitting here at the desk, reading some nonsense about A4V or JFK, it seems better done with an absolute poker face.
Did you read my link? What you have described is essentially the exact description given for tut-tut:
Sorry, edit changes preceded your reply. I think multiple clicks and a single click are slightly different expressions.
“At loss for words”?
The sound is technically called a dental click, which I have usually seen spelled tsk.
And this, according to George Carlin, is a bilabial fricative… 
Okay, so we have a technical term, and an associated term. But “Fred gave a dental click” really isn’t going to cut it, and tsk or tsk-tsk, besides being soooo Mauve Decade, seems to represent a different action and meaning.
To me, the double or triple click of tsking or tutting is a sub-verbalization of “no, no[, no]” - a civil if not polite way to emphasize a disagreement. “Fred waved his hand and tsked at me. ‘No, you don’t understand,’ he said.”
The dry single click is closer to the rude - “You’re shitting me, right?” - and it seems to deserve a specific conversational/narrative name of its own. After all, “Fred rolled his eyes at me,” is almost equivalent in meaning and easily understood.
I’d bet the !Kung have a word for it.
“Clucking your tongue” at someone.
By Jove, I think you’ve got it, old man. But I still think that implies multiple clicks, clucks or tsk/tuts. And still pretty… early 20th. Let’s make up a better word.
“Dry clicking” has nice nuances. 
Do you have an example on youtube or something? I’m trying to imagine how I’d click to express disbelief and I can’t picture it.
I’m not picturing it either. “Clucking the tongue” doesn’t really involve the cheeks, at least when I do it.
I just tried out the exact motions from your OP and get what you mean - it’s not clucking or tutting. It’s often accompanied by raising your eyes to the ceiling, but not by rolling your eyes, and often followed by a drawn-out “well…”
If that is what you mean, I don’t think there’s a word for it.
I heard a lot of “Tut Tut” from much older women when I was under 10 or so… But that had a different, to me, meaning than “Tisk Tisk”.
Tut = never you mind, mind your own business etc…
Tisk = That was naughty, against the implied or normal rules for time & place…