1. Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
2. In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
3. In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy, about 3-4 minutes.
4. Add the egg, maple syrup, and vanilla extract to the butter mixture, beating until well combined.
5. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
6. Drop tablespoon-sized scoops of dough onto the prepared baking sheet, spacing them about 2 inches apart.
7. Bake for 10-12 minutes, or until the edges are lightly golden. Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack.
8. To make the icing, melt the butter in a small saucepan over medium heat, stirring continuously until it turns a light brown color and emits a nutty aroma.
9. Remove from heat and whisk in the powdered sugar, milk, and vanilla extract until smooth.
10. Once the cookies are completely cool, drizzle or spread the brown butter icing over each one.

As you bake, savor the rich aromas that fill your kitchen, much like the ones that filled my childhood home.