“Tut” has been around probably as long as speech. It’s that sound you make with your tongue against the roof of your mouth, right behind your top front teeth, to express irritation or frustration. Mostly practiced by females. When I was a kid, I always had to have the last word, so if my dad would scold me, my final sound would be, “tut!”. To which my dad would exclaim, “Don’t you “tut” me! Actually, that’s the only time I’ve ever heard a straight guy use the expression.
Might have been better if she had tsked instead of tutted.