Wat is not a magi-priest, but cool that you brought him up. So on topic of Aavltovlekaii
The one Quarvlákaei are wearing is called Matama, tho it is of the same shape and design. Just as Aavltovleka it is plain white, however it has sewn in blue decorations, by tradition it should have two crescent moons on the spots where the wearer's "knees" are, and an axe, with its blade facing left side.
They wear it in this configuration, the round end in front, and it covers their eyes fully, this branch of Quarvlákei doesn't use eyes to orient themselves. However...
As the frog races undestand magic there are multiple ways one cast spells, the most common being by touch/movement (hands), voice (tongue), or sight (eyes). The tradition of Quarvlákei leaves their members to choose which one suits them best. Those who prefer doing magic with their sight would then expose their eyes when neccesarry.
So when you see a Quarvlák bending his Matama backwards and looking at you with his very eyes, he means bussines.
Judges wear their Aavltovlekaii like this - the round end is in the back. This still can cover their eyes slightly, though to not by far such extent as Quarvlákaei, even then, they can simply push the garment behind their head, so it doesn't affect their sight at all.
Regarding pins - the one frog (Dakva) you found on the First Twelve Kevekii pic was also the first judge of Agiii-Wuuvohuk and first judge period. Judges of Agiii-Wuuvohuk traditionally wear their Aavltovlekaii differently, with the left side pinned to right "shoulder", the pin has inscribed a number on it, so each Judge of Agiii-Wuuvohuk has his own.
With the models I did some tests on this, and came to pretty obvious conclusion that you can't pin normal Aavltovleka like this, one side of it needs to be prolonged considerably, so the shape would need to be different.
In any case, you would definitely need these cloak like garments to be held by some kind of string. Judges don't leap around much, but a string under the neck wouldn't hinder them at all, while improving the practicality of their clothes tenfold, so yeah, pretty obvious and intuitive.
Back to Wat, I imagine him wearing his shroud like this, the round end goes to the back, and the two sides aren't held by a long string, they're clipped very close to each other, making this shrouded, hooded kind of look.
In terms of lore, Wat is a character from a very old story of mine that is way ahead in the timeline, so while the character/personality/relationships are all okay, the setting he's in is very different from what I'm working on now and needs lot of fleshing out to keep it up to date.
Lastly, I wanted to visualise the pointiness of one's hat, because the one cap I did for the models isn't quite good enough to explain what I meant - Basically, you have some sort of basic headwear for the frog, in my case it was very simple, hood-like piece, that doesn't seem very practical for it covers the eyes. Not all frogs are wizards, so that's undesirable.
But then I made this curious discovery, that when you make the thing pointy, the pointier it is, the more can the frog see, while the overall shape of the hood/cap stays more or less the same, it just affects the pointy top and the area covering the eyes!
I remember some guy while back, Dunamisdeos probably, be super crazy about frogs and pointy hats, but man, POINTY HATS WERE THE WAY ALL ALONG!