The chaperone (or chaperon) was a popular hat style in the mid-15th century, especially in Burgundy. It's style was derived from the hood. There's an excellent illustration of this on The Costumer's Manifesto website, Figure 5. In addition, there are many period pictorial references on Wikipedia.

The black chaperone above is constructed of a melton weight felted wool, while the white chaperone is of flannel wool. The heaver, melton weight has a little more body to the fabric and works a little better for the oak leaf dags of the cockscomb or what on the hood would have been the mantle.

The roll is a rectangle sewed into a tube and stuffed with cotton padding. It's certainly unlikely that in period cotton would have been used. It is much more likely to have been filled with wool or perhaps even some sort of form or support as depicted in Paolo Uccello's Support for the cappuccio worn as a turban c. 1450 (see the Wikipedia link above). The tube is then sewed together. Both the liripipe and the cockscomb are secured to the inside of the roll.