What could the dough of this cookie be made of? After repeatedly eating what I consider the most delicious cookie I have ever eaten, my wife soon became frustrated trying to replicate it. So I am reaching out to gather ideas. This is the cookie, from a pastry shop. The filling is some kind of soft nougat, but the mysterious part is the cookie itself.
Here are some thoughts:

We assume that ingredients include (shortbread): flour, butter, and sugar
We tried icing sugar but it became too sweet
We tried milk with unclear success (?)
We tried eggs (white, yolk) but the taste did not match and it became too soft
The surface is a bit sticky
The taste is sweet and salty, a bit toasted/burnt which makes it so special

Unfortunately, I was not able to take a picture of a full cookie, so here is it half-eaten, as I could not control myself.
What could the dough be made of?
This is quite a shot in the dark with the minimal information, but I will give it a try.
My best guess is that you already have all the ingredients you need: butter, sugar and flour. What you are missing is probably the ratio. The dark edge with the crisp texture and the nutty flavor is very likely sugar that has self-caramelized in melting butter (similar to the tuille somebody mentioned in comments). Also, the image shows that the cookie is &quot;flowing&quot; - it was not stamped out from a rolled short crust, for example, it was a ball which melted in the oven. And it melted enough that it became very thin, with the edges flowing out to become thinner than the center.
I would suggest that you don't continue trying random recipes, but find a recipe for a cookie of this kind and follow it. You are most likely to find these recipes as recipes for malleable cookies for making e.g. edible ice cream bowls. I don't have my books here, so I can't post the recipe I have (you can come tonight(CET) into chat, if you want) but you can just search for this type of recipe online. Even if you don't find the perfect match, it will probably be a better starting point than random trying.
If you insist on trying yourself though, or cannot find one of these recipes, I would suggest you modify a German shortbread ratio (3-2-1 flour-butter-sugar). Make sure you are using butter, not a margarine or other substitutes, these have different spreading behavior when melting. And then simply reduce the flour substantially, possibly also increase the sugar, and see where that takes you.