By request, here’s the pattern I use for my fuzzy hats. The hats are plausibly period but not based on any real research. I did see what looks like a fuzzy hat on the head of a Bulgar in an image from Vat.gr.1613, “Menologium Basil ii” in image 345. Other images I’ve commented on are in this post.
Cut out the pattern. Cut out 5 in wool and 5 in linen. Sew together with a 1/4-1/2″ seam allowance. Add fur (or don’t, the pattern would work without fur). I used fur that I got at Pennsic years and years and years ago.
Options: I’ve lined it, you don’t have to. You can put the seams in between the layers. You can put the seams to the outside (and do seam treatments over it). You can embroider on the panels. You can add patches to it. You can add pilgrim badges to it. You can add a tassel and bead to the top/middle. You can extend the pattern another ~2″ at the bottom and make a fold up edge. Have fun.