I have made a leather hat based on the 1656 engraving by Paul Fürst, the only contemporary picture of a plague doctor that I have been able to find.
You can see that it has a wide brim that slopes downward, and a short crown that is wider at the top than the bottom.
Here is my finished hat made of black vegetable tanned cowhide with a deerskin sweatband. The biggest challenge in making such a hat is shaping the crown.
Traditionally hats that are wider at the top than the bottom are shaped on hat blocks that break down into several pieces. A single block could not be removed from the finished crown, but the multiple smaller pieces can be extracted one at a time. Above is pictured the crown and brim blocks, shown together.
I made the plague doctor hat block using this technique, but I took it one step further. Since any cracks between the individual hat block parts would transfer to the wet leather, I covered the hat block with a layer of silicone rubber. In this photo the pale blue silicone cover is sitting atop the pink hat block pieces.
Here is the five-part hat block turned upside down, and without the rubber cover.
Here you can see the five individual pieces. I made these out of urethane resin filled with air-filled glass micro-balloons to make it lighter in weight and easier to sand.
To assemble the hat block the silicone cover is inserted into the dampened leather crown (not shown). The front section of hat block is placed into the cover.
Followed by the back section.
Next the ride side.....
...and then the left side.
Finally the center section is put into place. Notice how the center section is tapered on all four sides, so that it is easy to remove.
When the damp leather dries it shrinks, and can tighten down on the hat block, thus the taper is an important element.
I have listed the plague doctor hat for sale at my Etsy store.