To begin, preheat your oven to 350 and line two baking pans with parchment paper (or spray with non-stick spray).
To a mixing bowl, add your dry ingredients: flour, cinnamon, nutmeg, baking soda and salt, and stir them together.
To another mixing bowl, add your butter and brown sugar and beat together with a hand mixer till creamy. Add your applesauce and vanilla and mix together. Add your egg and lightly mix it in.
Add your dry ingredients to your wet ingredients and stir them together till well combined.
Add your oats, apple, and walnuts and fold them into your batter. Let your batter rest for 10 minutes so the oats soak up the dry ingredients. This will ensure your cookies are thick and don't spread too thin.
Using a small cookie scoop, scoop the cookies onto your pans, evenly spacing them apart for 12 cookies per pan. Gently flatten the tops of the cookies (as these cookies don't spread much). Bake in the oven for 12-16 minutes, or till the tops are set.
To a small mixing bowl, add your powdered sugar and milk and whisk them together.
After your cookies have cooled, drizzle the glaze across the top. Devour!
Store these cookies in an airtight container, with parchment paper in between each layer of cookies so they don't stick together. They will last 1-2 days on the counter, and a week in the fridge.