Preheat the oven to 350°. Weigh the flour and set it aside. In a stand or handheld mixer, beat the butter until light and fluffy. Add the salt, vanilla extract, and pure maple, mixing through. Slowly add the flour and mix until all the flour is incorporated. Finally, add the chopped black walnuts.
I take a medium cookie scooper and add 12 scoops to the parchment-lined cookie sheet pan. I take a water glass with a flat bottom and gently press it on the top of each cookie dough scoop. Press a second time a little harder. If the cookie dough sticks to the glass, wipe it with a wet paper towel. You want the cookies to be about ½ inch thick. I do this to make it easier to put on the icing. These do not spread much, so I got 12 on my pan.
Bake for 11-13 minutes. When there is a little brown around the bottom fringe, take them out of the oven. Leave them on the pan for about 1 minute, then move them to a cooling rack. Let them cool completely before starting the brown sugar icing.
In a small saucepan, combine the milk and the light brown sugar (I weighed out), stirring over medium heat. I stir this almost constantly. You want to bring this to a boil. Turn the stove heat down just a little; you want to low boil for 3 minutes. You want the sugar and milk to thicken. Remove from the heat and add the butter, stirring until it melts.
Add 1 ½ cups of the sifted powdered sugar to a stand mixer or stir by hand with the brown sugar mixture. I like to do this in the stand mixer to get out all lumps. Blend till smooth. I mix the icing in the bowl for a couple of minutes to see how thick the icing gets. If the icing is thin, add a little of the ½ cup of powdered sugar you have left over from the 2 cups. Using a knife, add the icing to the top of each cookie. The icing hardens pretty fast.