Pour the apple cider into a small saucepan and heat it to a gentle boil over medium-high heat until it reduces down to ¼ cup, about 15-20 minutes.
In another saucepan, melt butter on low heat until it begins to foam; whisk frequently until it becomes fragrant and lightly browned.
In a separate bowl, whisk together flour, baking powder, baking soda, cornstarch, spices, and salt until combined.
In a large mixing bowl, combine cooled brown butter with sugars; then add egg, yolk, apple cider, and vanilla extract until well blended.
Gradually stir dry ingredients into wet ingredients until just combined; let dough rest while preheating oven.
Use a scoop to portion dough onto lined baking sheets; bake at 375°F for 9-11 minutes until edges are set.
Cool cookies briefly on sheets before transferring them to cooling racks while preparing coatings.
Whisk melted butter and apple cider together in one bowl; combine granulated sugar and cinnamon in another.
Brush each cookie with apple cider butter before rolling in cinnamon sugar mixture on both sides.
Mix powdered sugar with apple cider and vanilla extract for icing; drizzle over cooled cookies before serving.