Verb for the act of making a "Tch" sound

I would use "chuffed".

chuff n. 1. A sound of or like the exhaust of a steam engine. - v.i. 2. to emit or proceed with chuffs. The train chuffed along.

Webster's Encyclopedic Unabridged Dictionary of the English Language 1996 edition.

I had a professor who often used tut or tut-tut to express disapproval.

From Merriam Webster:

tut (interjection, verb)

used to express disapproval or disbelief

intransitive verb tutted, tutting.

E.g., Bob tutted at Mary's absurd suggestion.

"Tsked" is an acceptable way to spell the sound you're thinking of.

interjection

1. (used, often in quick repetition, as an exclamation of contempt, disdain, impatience, etc.)
2. for shame!

Source: Dictionary.com