What could the dough of this cookie 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.