From the 1200's to the 1500's it was quite popular for men in Europe to wear a toque, a hat with no brim at all.
These were particularly popular in France and eventually became used only in kitchens.
The toque blanche became standard uniform for chefs in France around the 1800's.
A famous French chef named Boucher was the first to insist that all chefs wore the tall, round, pleated, white hat in the kitchen for sanitary reasons.
I never knew a chef's hat had a special name.