Brown the butter, set into a bowl to make the cookie dough. Remove 1 tablespoon of browned butter and reserve for the glaze.
Toast the oats: set the oats in a large frying pan over medium low heat. When the oats are hot they will become fragrant, move them around the pan a bit just to ensure they are all heated evenly. Set aside to cool.
Add the salt, vanilla and cinnamon. Whisk well to combine.
Add the sugar to the bowl with the butter and whisk well then let cool for about 5 minutes. Whisk in the egg for at least a full 60 seconds. Add the flour and stir to combine. Stir in the oats.
Cover the bowl and leave it at room temperature for 2 hours or in the fridge for 1.
Preheat the oven to 350 F. Line two greased baking sheets with parchment paper.
Scoop the cookie dough onto the prepared pan in portions of 2 tablespoons each. Optional: dip the cookies head first into a bowl of rolled oats, this adds a little extra oatyness to the cookie.
Bake the cookies for about 12 minutes, until golden on the edges.
Let cool completely before making the glaze.
To make the glaze: the butter will probably have turned creamy, warm it for 5 seconds to re-melt it. Add a splash of vanilla, a pinch of salt and ¾ cups powdered sugar and 1-2 tablespoons of milk or heavy cream. Whisk until pourable, but thick (add more powdered sugar if it’s too thin, add more milk or cream if it’s too thick). Dip the completely cooled cookies face down in the glaze.