Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper.

Whisk together the flour, baking soda, and salt in a medium bowl until combined.

In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy. Beat in the egg, maple extract, and vanilla extract until well combined.

Gradually add the dry ingredients to the wet ingredients, mixing until just combined and a soft dough forms. Do not overmix.

Drop rounded tablespoons of dough onto the prepared baking sheet, spacing them about 2 inches apart. Bake for 8-10 minutes, or until the edges are golden and the centers are set. Let the cookies cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.

Melt the butter in a small saucepan over medium heat. Continue to cook, stirring occasionally, until the butter turns a golden brown color and smells nutty. Immediately remove from heat to prevent burning and let it cool slightly.

In a medium bowl, whisk together the powdered sugar, maple syrup, and cooled brown butter. Gradually add milk, one tablespoon at a time, until the icing reaches a smooth, spreadable consistency. Stir in the vanilla extract.

Once the cookies are completely cool, spread the brown butter icing over each cookie.