A similar style of hat is reffered to as a bonnet in English dress of this style, so I will use this term until I am sure of the German term. Mine is made from wool and lined in fake fur. It is slashed at the brim and only has one flap underneath as there are examples like this from the latter part of the century.
These can be worn over a haarhaub or simply worn on its own.
I have chosen to "bagline" each element (upper slashed circle for the crown and ear flap) then sew them both together. This saves a lot of hand work for me and still gives a neat appearance. It does allow the pile of fur to flow over the seams still as if it was rolled over the wool and hand sewn.

This is the large brimmed "pizza hats" reffered to in many reenactment circles. I decided to try for a construction I know was used in period rather than the very fabric consumptive "origami" crown as created by a reenactor several years ago (long before I got involved in any way shape or form of historical dress.) I also couldn't really think of a way to use much less fabric to generate the square crown seen in some images.
I decided to use a round crown pleated into the brim much as the extant barret does in the Nurenberg National Museum. This barret does not have a brim as such, but a boa of ostrich feathers instead.
It is essentially three large circles of wool. Two layers used for the brim each interlined and sewn over a hat wire (a witches hat from a halloween sale sufficed nicely for this.) The oval shape for the head was cut. The crown was lined in a cotton crinoline )I had thought I would interline it with this then line with another fabric, but I have not yet decided if this will happen, it might be best to leave the proper lining til last to reduce bult in the pleating. A ribbon will bind the hole once I decide on a good fabric that will not stain my skin.

This can be worn with or without a veil. Mine is based on a portrait of a woman by Wolf Traut and I have made the matching hemd.
The haube is made from essentially three parts: a shaped cap to fit closely to the head over the hair, the wuslt (crescent stuffed piece to create the shape) and the cover.
I made my fitted cap from a rectangle of fabric that fitted around my head. Then it was shaped in a way that is too complicated to describe when a diagram will do.
I tacked the wulst in position in three places: the centre of the wulst around the crown of my head, and the two points near the nape of my head. This is simply for expediency sake. It could just as easily be pinned into place each time I wore it.
There is as much variation in the covers of the wulst and cap as there is in the shape of the wulst. I chose a heavily gathered style.
I cut a semi circle of fabric, the flat edge encircling my head and the curved piece gathered and pleated in the back. This ensures a very tight fit to the head to stop it slipping around too much.
I also experimented with draping a veil over, but was never satisfied. I think a narrower veil would have been need than what I used. I did note how to get the square look seen in several 2nd quarter 16thC portraits. If you fit the veil over the haub and then pull gently on the fabric over the wulst itself at points directly behind your temple you can tweak the front edge just enough to get a rounded square shape. When you look at some portraits there is a heavy trimming to the edge of the weil. This may well help create this effect.

Seen in numerous Cranach paintings as well as paintings by other artists. There seem to be a micture of materials able to be used. I chose a metal tissue with an embroidered pattern that was at least similar to patterns seen in brocades of the time.
These differ a little to cauls seen all over the rest of Europe as these lie very flat to the front of the head and the bulk of their body is at the sides and bottom. This is rather easy to do to a degree;) I used an oval shape (like a sideways 0) and sewed it to a band easing the gathers to the sides and back rather than at the top. I have to pad my haarhaube to get the right shape.
This can be worn alone or combined with the bonnet or barrett.