Mix the Dry Ingredients: In a medium bowl, combine the all-purpose flour, salt, baking soda, and ground nutmeg. Whisk until evenly combined, then set aside.
Combine the Wet Ingredients: In a large mixing bowl, whisk the melted and cooled butter with the brown sugar until smooth. Add the egg, maple syrup, vanilla extract, and maple extract.
Whisk Until Smooth: Whisk the wet ingredients until everything is thoroughly combined and the mixture looks smooth.
Prepare and Chill the Dough: Add the dry ingredients to the wet mixture. Stir gently until just combined and no streaks of flour remain. Cover the bowl and refrigerate the dough for at least 30 minutes, ideally 1 hour.
Shape the Cookies: Roll the chilled dough into balls approximately 1½ inches in diameter. If desired, roll each ball in granulated sugar for a lightly crisp coating. For a softer, more textured cookie, skip the sugar. Arrange the dough balls on a parchment-lined baking sheet, leaving enough space between them to spread.
Bake the Cookies: Preheat the oven to 350°F (175°C). Bake for 8–10 minutes, or until the edges are just beginning to set and the cookies look slightly puffed. Avoid overbaking, as the cookies will continue to firm up while cooling.
Prepare the Maple Icing: Allow the cookies to cool completely. In a small bowl, whisk together the powdered sugar, maple syrup, and enough milk to achieve your preferred icing consistency.
Decorate and Serve: Drizzle the icing over the cooled cookies or dip each cookie into the icing for a fully coated finish. Allow the icing to set before serving. Enjoy your homemade Maple Brown Sugar Cookies!