Preheat oven to 350 F. Line two baking sheets with parchment paper.
Start by browning the butter. You'll take 10 tablespoons of butter and melt in a medium saucepan, and then continue to cook past the point of boiling. You will do this over medium heat, past the point of the butter foaming, until it turns a nice golden brown, and has a nutty smell. Make sure you are stirring and watching it, because it will turn from perfectly brown to burnt fairly quickly. Once finished browning, pour into a cup or bowl and remove 1 tablespoon to be used for the glaze.
Peel and dice up your apple very small and mix together with 1 tablespoon of brown sugar and ½ tablespoon of cinnamon. Set aside.
Place all but 1 tablespoon of browned butter and both sugars in a large bowl and mix together for 1-2 minutes. Add in the egg and vanilla and mix again.
Mix in the salt, baking powder, baking soda, cinnamon, nutmeg and flour until combined.
Fold in your apples. If there is liquid in the bottom of the bowl do not add that in.
Scoop the dough into 3 tablespoon size balls and place 6-7 per baking sheet.
Bake the cookies for 10-12 minutes. You're looking for a light golden brown outside rim.
Mix together the brown butter, powdered sugar and milk for the glaze. Brush the cookies with the glaze while they are still slightly warm, but not hot, and sprinkle with flaked salt.