Preheat the oven to 350°F and line 2 baking sheets with parchment paper.
Brown the butter by simmering on low until brown flecks appear. Remove from heat and allow to cool for 5 minutes.
Pulse the old-fashioned oats a few times in a food processor or blender until roughly chopped. Do not overpulse into oat flour, this will ruin the texture of the cookies.
In a small bowl, whisk together roughly chopped oats, all-purpose flour, baking powder, baking soda, and cinnamon. Set aside.
In a medium bowl with a hand mixer on medium speed or in the bowl of a stand mixer, mix the browned butter, brown sugar, and white sugar. Add in the egg yolks, sourdough starter, and vanilla, and mix well.
Add the dry ingredients to the wet ingredients and mix until incorporated, do not over-mix or the cookies will be tough. Add the cinnamon chips and stir into the dough.
Use a cookie scoop or spoon the dough onto a parchment paper prepared baking sheet and use the back of a spoon to flatten the dough to 1/4” thick circles, or roll the dough into balls and flatten it before baking.
Bake at 350°F for 10 minutes or until the edges are golden brown.
Set the cookies on a wire cooling rack to cool completely before icing.
In a small bowl, whisk powdered sugar and milk to make a thick icing that runs off of the whisk in ribbons.
Quickly dip each cooled cookie into the icing face down and use a fork to lift it if necessary to prevent breaking.
Place the cookies back on the wire rack with parchment paper underneath to catch drippings. Allow the icing to set and dry for 15 minutes.
