Step 1: Preheat oven to 375°F (190°C). Line baking sheets with parchment paper.
Step 2: In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
Step 3: In a large bowl, cream together the softened butter, granulated sugar, and brown sugar with an electric mixer until light and fluffy.
Step 4: Beat in the egg, then stir in the maple syrup, vanilla extract, and optional maple extract until well combined.
Step 5: Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Step 6: Drop rounded tablespoons of dough onto the prepared baking sheets, about 2 inches apart.
Step 7: Bake for 10-12 minutes, or until the edges are lightly golden and the centers are still slightly soft. Let cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
Step 8: While cookies cool, prepare the brown butter icing. Melt 1/2 cup unsalted butter in a light-colored saucepan over medium heat. Cook, stirring frequently, until the butter turns a golden brown color and smells nutty. Remove from heat immediately and pour into a heatproof bowl to cool slightly.
Step 9: Once the brown butter has cooled (but is still liquid), whisk in the powdered sugar, milk/cream (start with 3 tablespoons and add more if needed for desired consistency), vanilla extract, and a pinch of salt until smooth.
Step 10: Once cookies are completely cool, spread or drizzle the brown butter icing over each cookie. Let the icing set before storing.