There is a fairly well known recipe for Triple Cooked Chips. It involves cooking chips three different ways / times.
Is triple cooked correct or should it be called Thrice Cooked Chips as they have been cooked three times?
Pretend it would concern the number two instead of three:
Use thrice when you would use twice.
Use triple when you would use double.
As they have been cooked three times, I would suggest "thrice".
For comparison: I have seen you triple today. Doesn't work, does it?
If it is a fairly well-known recipe, then the name is a fixed phrase by now — and also, well, a name. So it cannot be wrong by definition. It's a label, and as far as the English language is concerned, it can be anything at all. You might as well wonder if New York should be called Second York instead. Or Little Pear. It should nothing. It just is.
Thrice is rather old fashioned these days, even archaic. It's valid, but 'triple' conveys the meaning perfectly adequately.
As other people point out, many people consider triple to be a modern 'replacement' for thrice, and in advertising, what matters is your customer's opinions, not grammar nerds on the internet.
That said, thrice is more specific and accurate in this case.
"I had coffee twice/thrice" involves a lot more washing up than having a double coffee or a triple coffee.
So triple cooked chips could simply be cooked for triple the amount of time, at triple the 'normal' temperature, or they could be cooked three separate times.
Thrice cooked chips, however, have to be cooked three separate times. It cannot be one longer/hotter cook.
Using Thrice is more descriptive, and in order to be completely understood, you should use thrice. If you want to sell chips, do some market surveys of your target audience and go with the majority.
"Thrice cooked" is a bit archaic. Correct English would be "Triply cooked" in my opinion. "Triple cooked cookies" rather sounds like triple cookies rather than triply cooked.