Cream the Base: In a large bowl, beat the softened butter, brown sugar, and white sugar until light and fluffy (about 3 minutes).
Add Wet Ingredients: Add the egg, ⅓ cup maple syrup, vanilla, and maple extract. Beat until smooth.
Add Dry Ingredients: Whisk the flour, cornstarch, baking soda, and salt together. Gradually add to the wet mixture on low speed. Mix just until no flour streaks remain.
Chill: Cover the dough and refrigerate for 1 hour. This is crucial to prevent spreading and keep them thick.
Roll and Bake: Preheat oven to 350°F (175°C). Scoop rounded tablespoon-sized balls of dough onto a parchment-lined baking sheet. Bake for 10–12 minutes. The tops should look set, but soft. Let cool completely on the pan.
Brown the Butter (For Icing): Place the ¼ cup butter in a small saucepan over medium heat. Cook, stirring constantly, until it foams and turns a golden amber color with a nutty aroma (about 5 minutes). Watch closely so it doesn't burn! Remove from heat immediately.
Make the Icing: Pour the browned butter (including the brown specks!) into a bowl. Whisk in the powdered sugar, 2 tbsp maple syrup, and enough milk to make a smooth, spreadable glaze.
Dip: Spoon the warm icing over the cooled cookies, or dip the tops directly into the bowl. Let set for 20 minutes.
