Isis from Medieval Silkwork can tell you a lot more about it - here is just a quick glimps on my new frilled veil.
As the process of weaving a veil with a frilled border is really slow and I don't have much time to weave at the moment, I got a bit impatient. I just had to make something I could wear now, not next year...
It's a simple rectangel of handwoven linnen cloth bordered with a narrow linnen band. The band has two different edges: the straight edge is sewn to the cloth, the other edge is curved and forms the frill.
There are some pictures of this type of frill painted by Conrad von Soest (e.g. on the altar of the Stadtkirche in Bad Wildungen, painted around 1403).
You can even see the different colour of the frilled border on the lower edge on Marys veil:
On another painting you see a tiny line above the frill - so I think this could be some kind of band sewn to the cloth...
I got lucky and found a border of handwoven linnen with one straight and one curved edge on a piece of vintage linnen from the 19th century that I could use for my veil :-)
Technically not very difficult and quite similar to the woven garter with a dagged edge from 14th century London...
Unfortunately I don't have a good photo of me wearing the veil in this style at the moment, just one taken from the side while I was explaining something to a bunch of visitors...