Preheat the oven to 375 degrees. Line a baking sheet with parchment paper.
Measure out one the walnuts and let toast for 7-8mins. Chop and set aside.
Melt butter in a skillet or heavy bottom medium-sized pot over medium-high heat.
Swirl the butter in the pan or pot as it begins to boil. The butter should boil out the frothy phase and begin to turn an amber brown color. Remove, add to a bowl and let cool for 15 mins.
In a large bowl add chopped walnuts, flour, cinnamon, baking soda and salt.
In a small-medium sized bowl combine the egg yolk, applesauce, vanilla and brown sugar with a whisk and add in the cooled brown butter (Strain it if you have a lot of larger brown butter particles)
Combine the egg and butter mixture to the dry ingredients and stir with a spatula or wooden spoon until well incorporated.
Using an ice cream scoop or rounded spoon, drop cookie dough onto parchment covered baking sheet.
Bake for 10-12 minutes. The edges should be golden brown and the center firm.
Allow cookies to cool on baking sheets for 10 mins before removing them to a baking rack for icing.
Make the glaze by combining the powdered sugar and milk in a bowl.
Stir until there are no lumps.
Using a spoon drizzle icing over cookies and let sit for about 20 minutes.